Lectures in Rings and Semigroups

Lecture Notes in Mathematics Edited by A. Dold, Heidelberg and B. Eckmann, Z~Jrich 380 Mario Petrich Pennsylvania State...

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Lecture Notes in Mathematics Edited by A. Dold, Heidelberg and B. Eckmann, Z~Jrich

380 Mario Petrich Pennsylvania State University, University Park, PA/USA

With an Appendix by Richard Wiegandt

Rings and Semigroups

Springer-Verlag Berlin • Heidelberg • New York 19 74

AMS Subject Classifications (1970): 20-02, 20-M-20, 20-M-25, 20-M-30, 22-A-30, 16-02, 16A12, 16A20, 16A42, 16A56, 16A64, 16A80

ISBN 3-540-06730-2 Springer-Verlag Berlin • Heidelberg • New York ISBN 0-387-06730-2 Springer-Verlag New Y o r k . Heidelberg • Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. © by Springer-Verlag Berlin • Heidelberg 1974. Library of Congress Catalog Card Number 74-2861. Printed in Germany. Offsetdruck: Julius Beltz, Hemsbach/Bergstr.

INTRODUCTION

Semigroup theory can be considered as one of the more successful offsprings of ring theory.

The relationship of these two theories has been a subject of particu-

lar attention only w i t h i n the last two decades and has generally taken the form of an investigation of the m u l t i p l i c a t i v e semigroups of rings.

The first and still

the most fundamental w o r k in this direction is due to L.M. G l u s k i n who certain dense rings of linear transformations These Lectures

represent an attempt

from the m u l t i p l i c a t i v e

studied point of view.

to put selected topics concerning both rings of

linear transformations and abstract rings, as well as their m u l t i p l i c a t i v e semigroups,

into a form suitable

for presentation

to students interested in algebra.

The Lectures are divided into three parts according to the clusters of covered topics.

part I consists of a study of certain semigroups and rings of linear transformations on an arbitrary vector space over a division ring. linear transformations two phenomena,

For dense rings of

containing a nonzero linear t r a n s f o r m a t i o n of finite rank,

from the present point of view,

are of decisive importance:

(a) its

m u l t i p l i c a t i v e semigroup is a dense extension of a completely 0-simple semigroup, and

(b) it has unique addition.

can be obtained by considering naturally

Because of (b), most

their m u l t i p l i c a t i v e

information about these rings

semigroups alone.

This leads

to a study of semigroups of linear transformations and in particular of

those satisfying

(a) above.

Hence in many instances we first estab]ish the desired

result for semigroups and then specialize

it to rings of linear transformations.

Even though the guiding idea adopted here is that first expounded by Gluskin the principal references are the books of Baer

[i] and J a c o b s o n

part II contains an investigation of various abstract rings, izations and representations.

[4],

[7]. their character-

The classes of rings under study here are semiprime

rings with minimal o n e - s i d e d ideals subject to various other restrictions. each of these classes of rings a m u l t i p l i c a t i v e b e i n g possible in view of their unique addition.

characterization

For

is provided,

this

Their m u l t i p l i c a t i v e semigroups

are either dense extensions of completely 0-simple semigroups or of their orthogonal sums.

Again some of the basic ideas here stem from G l u s k i n

reference is Jacobson

[4], but the m a i n

[7].

part III represents

a topological treatment of a left vector space, and its

ring of linear transformations, topology of modules and rings,

in duality with a right vector space.

Linear

linear compactness of vector spaces, various

IV

topologies on certsin rings of linear transformations, s topological vector space are covered here. appear here as topological rings. are due to Dieudonn~ and K~the

as well as s completion of

C e r t a i n of the rings studied earlier

Many of the basic

[2], but the chief references

ideas discussed in this part

sre the books of J a c o b s o n

[7]

[i].

The appendix contains s concise exposition of some principal achievements in the theory of linesrly compact modules and semisimple rings. consist of several characterizstions rings.

The chief original

The two main results

of linesrly compact primitive and semisimple

reference here is Leptin

12}.

Jacobson's Density Theorem

is included with a short proof.

The m e t h o d of m a x i m u m exploitstion of the m u l t i p l i c a t i v e often mskes

the use of various hypotheses more

that, st least for the rings under study here,

structure of a ring

transparent and demonstrates

the fact

the sddition is essentially extrane-

ous. The sources of these Lectures are numerous: references,

in addition to the above m e n t i o n e d

they include s vsriety of papers which sre generslly referred to in the

text.

There is slso a generous

time.

Among these are the results concerning the structure of simple rings with

minimal one-sided

sprinkling of results published here for the first

ideals in terms of Rees matrix rings,

and some related results due to Dr. E. Hotzel; author.

isomorphisms of the latter,

the remsining ones are due to the

For the sake of clarity and u n i f o r m i t y of presentstion,

have been rephrased and several new proofs hsve been provided. exercises at the end of most sections;

many known results There are several

they are designed to test the u n d e r s t a n d i n g

of the m a t e r i a l and sometimes extend the subject covered in the text. msterial

However,

the

in the msin body of the text is independent of exercises.

Part I was the subject of a one semester course in linear algebra in the Summer of 1969;

the entire Lectures

formed the content of a two semester course in topics

in ring theory in the school year 1971/72, both at the Pennsylvania State University. I sm indebted to Dr. D.E. Zitsrelli

for taking the notes

for Part I, to

Mr. JoJ. S t r e i l e i n for taking the notes for parts II and III, supplying the appendix,

and to Professor B.M. Schein

suggesting several improvements.

I am grateful

to include his u n p u b l i s h e d results, many slips,

to Dr. R. W i e g a n d t

for

for reading the m s n u s c r i p t and

to Dr. E. Hotzel

for the p e r m i s s i o n

to students in the two classes for correcting

as well as to all other persons who c o n t r i b u t e d

to the existence of these

Lectures. Statements in the text are referred to only by number: the same part,

the Arabic numerals are used,

if the statement is in

say 5.6 which is statement

6 in

V

Section 5; if the statement is in a different Part, numeral in affixed,

the number of the Part in Roman

say 1.5.6.

Since the w o r k of G l u s k i n on the subject at hand has provoked considerable interest,

it is hoped that a systematic and s e l f - c o n t a i n e d e x p o s i t i o n will propa-

gate the existing knowledge and stimulate new research in this highly promissing area of the r i n g - s e m i g r o u p cooperation.

TABLE OF CONTENTS

PART I SEMIGROUPS AND RINGS OF LINEAR TRANSFORMATIONS I.i

Definitions and notation

l

1.2

Dense rings of linear transformations

1.3

One-sided ideals of

1,4

Ideals of

1.5

Semilinear isomorphisms

34

1.6

Groups of semilinear automorphisms

47

1.7

Extensions of semigroups and rings

54

g(V)

Su(V)

7 16

and principal factors of ~ u ( V )

27

PART II SEMIPRIME RINGS WITH MINIMAL ONE-SIDED IDEALS II.l

prime rings

11,2

Simple rings

74

11.3

Maximal prime rings

85

11.4

Semiprime rings

91

65

95

ii .5

Semiprime rings essential extensions of their socles

11.6

Semiprime atomic rings

102

11.7

Isomorphisms

105 PART III

LINEARLY TOPOLOGIZED VECTOR SPACES AND RINGS III.I

A topology for a vector space

112

III.2

Topological properties of subspaces

120

III.3

Topological properties of semilinear transformations

125

111,4

Completion of a vector space

128

111.5

Linearly compact vector spaces

131

111.6

A topology for

£u(V)

136

III.7

A topology for

Su(V)

139

111.8

Another topology for

III.9

Complete primitive rings

£u(V)

143 146

VIII

APPENDIX ON LINEARLY COMPACT PRIMITIVE AND SEMISIMPLE RINGS O.

Introduction

152

i.

More about primitive rings

153

2.

Inverse limits and linearly compact modules

156

3.

Linearly compsct primitive rings

159

4.

Linearly compact semisimple rings

162

CURRENT ACTIVITY

167

BIBLIOGRAPHY

168

LIST OF SYMBOLS

175

INDEX

177

PART I

SEMIGROUPS The subject transformations dense

rings

finite mations one,

of

this part

rank h a v i n g

all o n e - s i d e d

vector

transformations

are c h a r a c t e r i z e d

of finite

ideals

are

in a vector

found,

the semigroup

taining

all

linear

duality w i t h

transformations

the given one

the u n d e r l y i n g vector

space

vector

of rank

is e x p r e s s e d

spaces.

are discussed.

in semigroups

representation

two semigroups

and rings with

part

space

of

In particular,

i with

linear

transforthe given

set of idempotents factors

an adjoint

is con-

each

are

con-

space

in

transformations

of

automorphisms

on a

of semilinear

of ideal

semigroups

is

ideals

in a vector

a discussion

to certain

all

transformations

of semilinear

of groups

of

in duality with

transformations

linear

of linear

transformation

for its principal

ends with

application

ring.

linear

ordered

linear

by means

A number This

and rings

For the ring of all

its partially

and ring of all

between

a division a nonzero

ways.

and a Rees m a t r i x

Isomorphisms

space over containing

semigroups

in several

characterized,

found.

TRANSFORMATIONS

an adjoint

structed.

For

OF L I N E A R

is a study of certain

on an a r b i t r a r y

of linear

rank

AND RINGS

extensions

and rings

of linear

transformations. The emphasis Thus many here

here

statements

for semigroups

is one

which

the m u l t i p l i c a t i v e

are usually

and are

then s p e c i a l i z e d

I.i A division all

the axioms

a commutative by

4.

The

ring

division

symbols

Throughout Denote

+,

this

the elements

case Roman

(or a skew

for a field

letters

in such a way

pair

that

is a field have

let

L

by lower

v,x,y, ....

is a function m a p p i n g The ordered

~

possibly

~X V (&,V)

under first

study. proved

AND N O T A T I O N is an algebraic

commutativi~y

system

satisfying

of m u l t i p l i c a t i o n .

and conversely.)

be a d i v i s i o n case G r e e k that

It will

V, say

postulates

ring and

letters, ~

be usually

(Thus denoted

V

be an a b e l i a n

and those of

acts on

V

V

by

group.

lower

on the left if there

(o,x) ~ ox.

is a left vector

the following

are

their usual meaning.

We say

into

of the rings

for these rings

to rings.

field or a sfield)

-, i, 0

section, of

DEFINITIONS

except

ring

structure

established

space

if

&

are fulfilled:

acts on

V

on the left

O ( x + y) = o x + o y

(oE 4, × , y E

v)

(O+~)x

(o,~ E a, x E

V)

~(~x)

= Ox+~x

= (o~)x

(x ~ V).

iX = x

The action and

those of If

~

is called

V

acts on

V

on the right,

postulates,

then

case,

using

the n o t a t i o n

Hence

the name v e c t o r

We will w r i t e

"V

Let called

the ordered

V

(left or right) the phrase

scalar m u l t i p l i c a t i o n ,

(V,4)

space

instead vector

pair

(&,V)

and

makes

of

(~,V)

(4,U)

linear

(~,V)

&

called

form

The set of all to be an abelian

linear

x(f+g)

group

V The

set

as follows. V*

V*

on the right

function

addition

V*

group

U

(&,V).

(V*,&)

forms on

for

is a right vector

Elements

of

V*

of confusion.

is to be regarded

group).

In such

A function

ss a

a case

a:V ~ U

is

V). spaces

apace

as operators

(&,A+) ~x of

of

(~,V)

where

on the right

on

into

4+

signifies

is the m u l t i p l i c a t i o n (4,V)

or functional)

into

(&,~+)

(&,U)

is easily

seen

by

(&,V)

f E V*

(i)

the usual

group

addition

can be made under

and

into

addition

o E &,

fo

of

the a b e l i a n

of homomorphisms. a right v e c t o r

(i).

We make

be defined

g

space act on

by

( x E v). space,

called

will be denoted

by

is

(~,V).

group of all h o m o m o r p h i s m s

under

×(fo) = (xf)o Then

act on the right.

danger

(x ~ V).

is an abelian

by letting,

In this

a,b,c, ....

defined

of the abelian

of linear

sp~ce.

if

transformation

= xf+xg

the abelian

First

A linear

transformations

under

it is a subgroup into

of left vector letters

as a left vector

(or a linear

group

(~,U)

and the scalar m u l t i p l i c a t i o n

o E 4, x E g+).

a linear

In fact,

itself &

V

spaces.

into

of the above

(x,y E V)

case Roman

g

group of

(here

scalars

4" is customary.

(o E 4, x E

transformations

lower

space over

be two left vector of

that

than just an abelisn vector

= xa+ys

We can consider

in

are called

a right vector

cases without

if it is clear

(rather

transformation

them by

the additive

A

that the scalars

in most

(ox)a = o(xa) We write

of

the right v e r s i o n

is called

it clear

can be used

space

(x+y)a

satisfying

(V,&)

is a (left or right)

a linear

and denote

elements

are called vectors.

the dual f,g,h,....

(or conjugate)

space of

3 A semigroup (usually

is a n o n e m p t y

mations

of

sition

(ab)x = a(bx)

written

on the right,

resulting

X

into itself,

semigroup

A linear (&,V).

written

(x C X),

on the left,

to be denoted

the c o m p o s i t i o n

will be denoted

transformation

of

g(V).

The same

the rin$ o f e n d o m o r p h i s m s

(&,V)

into

of

(~,V)

X,

binary

is a s e m i g r o u p

by

g(X).

operation

the set of all transforunder

the compo-

If the t r a n s f o r m a t i o n s

x(ab)

= (xa)b

(x E V)

and

are the

~(X). itself

(g,V) of

is called

of

~I(V);

and will be denoted

the addition

and will

an e n d o m o r p h i s m

is s s u b s e m i g r o u p

(g,V),

set endowed with of

set

is given by

by

the > e m i g r o u p o f e n d o m o r p h i s m s

or simply

an associative

For any nonempty

The set of all e n d o m o r p h i s m s

is called

£(V)

set together w i t h

called m u l t i p l i c a t i o n ) .

(i) forms

be denoted

by

by

a ring,

£(&,V)

of it

g(&,V) called

or simply

.

We have

seen that

can be given can be made made

If

U

transformations

group.

space over

4,

of

In addition, and for

(~,V)

into (~,U) + U = ~ , this set

for

U = V

the same

set can be

or a ring.

is a subgroup

is called

linear

of an abelian

into a right vector

into a semigroup

(&,U)

the set of all

the structure

of

a subspace

V

closed

of

(g,V).

under m u l t i p l i c a t i o n It is clear

that

by scalars

(~,U)

in

4,

then

is itself a vector

space. As in the case of a finite a linearly

independent

a generating and a basis

set

(every v e c t o r

of a vector

as a m i n i m a l

Any

have

two bases

denoted

to a basis

of

dim V.

every basis

To every

every vector

in

a complement

properties require proofs If mation

V

A

of a v e c t o r

further

V of

ranse of

and V

of

of

into

are U,

methods

which

then

in the form

V, and we write express

reference.

over

Va = {xa Ix E VJ

V

can be extracted.

to a basis of with

Most Zorn's

a

of the whole V

such

that

x E A, y E B;

We will

use

these

of these statements lemma).

For

is a linear

is a subspace

a, d e n o t e d

of

by rank a.

the null

V,

can be completed

the

at the end of this section. 4, and

called

of

V = A~B.

(axiom of choice,

spaces

the d i m e n s i o n

B

,

it is set

a basis

x+y

..

independent

set of vectors

a subspace

independent),

combination

has a basis;

linearly

can be completed

written

of

linear

set of vectors

one defines

is linearly

is by d e f i n i t i o n

there exists

is the rank of

is a subspace

always

independent

see the references

left vector

a; its d i m e n s i o n

N a = ix E V I xa = 0}

in

space w i t h o u t

discussion

U

A

subset

as a finite

space

generating

V,

space over a field,

finite

set or a maximal

linearly

can be uniquely

proof by transfinite and

A vector

of a subspace

subspace

B

is called

Any

V, and from every

In particular, space.

V.

generating

the same c a r d i n a l i t y

by

vector

(every

can be w r i t t e n

space

characterized

usually

dimensional

set of vectors

transfor-

U, called The set

space of

a.

the

Linear written space;

the r a n g e

space

are

a given

a function

left;

the null

vector

the dual

space

of a l i n e a r

pair

~:VX

(U,g)

U ~ ~

and

(&,V)

is c a l l e d

are d e f i n e d vector

=

(Vl,U)

+

(v2,u),

(v,ul+u2)

=

(V,Ul)

+

(v,u2) ~

~(v, u) =

of a r i g h t

and a left v e c t o r

form

u , u l , u 2 t U, o ~ &,

form

is n o n d e g e n e r a t e

and

(v,u)

stands

= 0

for all

u E

U

implies

that

v = 0,

vi)

(v,u)

= 0

for all

v 6 V

implies

that

u = 0.

In such

a case we

say

the d i v i s i o n

variously sizing

use

ring

Property (v,u)

is m o r e Let

over

(U,&,V)

LEMMA°

or

we

where

Oil

vector

should

~(v,u).

spaces

write

according

assumed

(U,&,V;~);

to the n e e d

as given).

is a left

(or a dual

Note

and

~l

that

pair)

we w i l l for emphain J a c o b s o n

is a r i g h t

vector

to: v E V

implies

for a p p l i c a t i o n ;

a vector

of dual

(U,V)

tacitly

for all

space.

a t-subspace)

If

by

t-subspace PROOF. is left

that

u = u ~,

analogously

A subspace

U

if for e v e r y

for v). of

V*

is c a l l e d

0 ~ x E V,

there

a total

exists

f ~ U

i__s_s!

pair,

vf u = (v,u) (V*,g).

t->ubspace

form then

(v,f)

the

of

function

(v E V),

(V*,g),

= vf

(v E V,

then f E

f:u ~ f u

is a l i n e a r

(U,V)

U).

(u E U),

isomorphism

is a dual

Conversely, where

of

pair

if

f u :V ~ &

(U,&)

onto

-is -

a

of

The

proof

consists

of a s i m p l e

application

of r e l e v a n t

definitions

as an e x e r c i s e .

If c o n v e n i e n t , as a t - s u b s p a c e U.

U

the b i l i n e a r

is a dual

fU = nat

is

is u s e d

convenient

with

defined

(~

(v,u ~)

be

precisely,

(U,&,V)

for

xf # 0

l.l.l &

=

is a pair

is e q u i v a l e n t

(or b r i e f l y

that

More

(~,~) vi)

(&,V)

subspace

(U,V)

4.

ring

the n o t a t i o n

vi ~)

that

the n o t a t i o n

the d i v i s i o n

space.

we m a y

space

if

(v,u)

over

vector

if it s a t i s f i e s :

v)

which

on a right

vector

(v,u)~ = (v,u~),

A bilinear

[7],

and

is a left

(ov,u) ,

V , V l , V 2 ~ V,

over

analogously

space

transformation

a bilinear

(Vl+V2,U)

where

spaces

of a right

analogously.

i)

iv)

and

of right

the

ii) iii)

such

on

and

defined

For 4,

transformations

as o p e r a t o r s

of

A dual

consider

V

we w i l l V*;

we

identify call

discussion

fU

u

fu' w h i c h

the n a t u r a l

is v a l i d

as a t - s u b s p a c e

and

of

imsse

by i n t e r c h a n g i n g U*.

amounts of

U

to c o n s i d e r i n g

in

the roles

V* of

U

and w r i t e U

and

V,

so

For a given of

a

in

U

(U,A,V)

(and

a

(v,bu) I.i.2 LEMMA. U,

if so denote

subrin$ o f

:

For a fixed b~u.

Then

b

and

=

(v,bu)

u, we o b t a i n Since

just

b~

V)

that

b

is an ad~oint

if

(2)

introduced,

a

has at most

~:a -- a*

one adjoint i n

is an isomorphism

of a

£(U,t~).

are adjoints

= (v,b'u)

(v,bu) =

u E U

in

the function

onto a subrin$ o f

If both

b

we say

(u E U, v E V).

the n o t a t i o n a*.

b E ~(U),

of

(va,u)

With

(va,u)

bu=

a E ~(V),

is an adjoint

it by

£(A~V)

PROOF.

and

(v,b~u)

is arbitrary,

of

a

in

U,

(v E V,

u E U).

for all

v ~ V

we have

then

(2) yields

which

b = b s, which

implies

proves

that

the uniqueness

of the adjoint. Suppose

that

s

has an adjoint.

Let

c~,~ E A, x,y E V;

then

for any

u E U,

we get

((ox+~y)a,u)

=

(ox+~y,a*u)

= o(x,a*u)

= o(×a,u)+T(ya,u) so that

(Ox+~y)a

Reversing the unique

= o(xa)

the roles

adjoint

Suppose

of

that

+ T(ya),

and

= a

of the adjoints,

a*

in

have

(~(×a)+T(ya),u)

is linear. we see

V, and that

a,c E 3~(V)

+ ~(y,a*u)

a*

adjoints

that

this

also

shows

that

a

is

is linear in

U.

Then

for any

v E V, u ~ U,

we obtain

(v,(a*+c*)u)

= (v,a*u+c*u) = (va,u)

which

implies

that

a*+c*

(v,(a*e*)u) which

implies

have

an adjoint

onto a subring on

U,

then

a

of

U

forms

£(U,g).

must

1.1.3 LEMMA.

For

i)

rank

(ab)<

min{ranka,

ii)

rank

(a+b)

< ranks

PROOF.

Va, then

Ab

i) Since generates

vab ~ Vab

of

By uniqueness function

a,b E £(V),

= (v(a+c),u)

Further,

=

It follows

a subring

be the zero

= (va+vc,u)

= (va,e*u)

a^c :" = (ac)*.

in

= (v,a*u) + (v,c*u)

(vc,u)

= (a + c ) * .

= (v,a*(c"u))"

that

+

((va)c,u) that

£(B,V)

= (v(ac),u)

the set of all and

~

of adjoints,

if

on

V.

Thus

~

a E $~(V)

is a h o m o m o r p h i s m a*

is the zero

which of it

function

is an isomorphism.

we have

rankb},

+ rankb. vb, we have which

rank

implies

(ac) ~ rank b.

that

If

A

is a basis of

rank

ii)

Let

generates

A

and

V(a + b )

Let then

(ab) = dim Vab < dim Va = rank a. B

and the desired

~(a,V)

~(&,V)

(U,g,V),

Then

A U B

follows.

£(&,V).

= ~a 6 ~(V) I a = ~a E ~ ( V ) f r a n k

has an adjoint

inherited

= ~(b,V)

a

in

U},

is finite],

from

£(~,V).

Note that

fL £U(&,V).

let = [a ~ ~(A,V)

n = 1,2,3,...,

the semigroup

We will omit ring.

h

Note

Ia

has an adjoint

in

U],

let

Sn,U(g,V)

division

Vb. respectively.

has finite rank];

SU(&,V)

~U(&,V)

both with

la

~U(~,V)

~U(&,V)

and for

and

let

the ring structure

Further

Va

inequality

= {a ~ ~(~,V)

is a subring of

For a given

both with

be bases of

= ta E £U(b,V) structure

I rank a < n],

inherited

from this notation that 1.3 implies

from

~(h,V).

if there is no need for stressing

that

£u(V)

is a ring,

and

~u(V)

the is a

semigroup. A nonempty

subset

b ~ I, we have and

~n,u(V)

PROOF.

is an ideal of

~u(V).

£v,(V)

It suffices

For every

Then

f E V*,

bf:V ~ &

follows easily

of a semigroup It follows

1.1.4 LEMMA.

V*.

I

ab,ba ~ I.

to show that every element let

bf

and is linear since both (f E V*)

The adjoint of

a E £(V)

in

a* a

the adjoint of

S

~u(V)

a

a

in

be the function given by

b:f ~ bf

of

is an ideal of

if for any

is an ideal of

a E S, ~u(V)

= ~(V).

that

by

S

from 1.3 that

V*

in any

a

and

f

are.

is the adjoint of

£(V)

has an adjoint

x(bf) = (xa)f Hence

bf ~ V*

a

in

V*.

is called the conjugate

of

a.

U, if it exists,

and in particular

and it

We will denote the conjugate

(which should cause no confusion). I.i.5 LEMMA. £u(V) PROOF.

Let

U

be a t-subspace o f

= is ~ ~(V) I a*U ~ U For

a E ~u(V)

let

where b

a*

V*;

then

i__~s th___~econjugate o f

be its adjoint

in

U.

in

(x ~ V).

a}.

Then for any

u E U,

(va,u)

so that

= (v,bu)

a*u = bu E U that

observe

Hence

a*IU

for

and

(U,~,V),

A

and

B

function on a set values

1

or

multiplicative with zero

if

semigroup

of

~.

references

1.2

If

(U,A,V).

Let

for each

i E IU, let

ui

in

Note

[i,0,~]

VX

For

If

The Kronecker ~

is a ring, ~

For

of

~

[i,~,k]*

[i,x,X]*

ring, ~

space)

means

the

([i], Chapter

II), Jaeobson

([1],§§7,8,9).

subspsces

in

Ui;

spaces

Ui

let

IV

of

U, and

be an index

~ ~ IV, let

v~

be a

the transformation

(v E V),

in

V for

(1)

for any

i E I U, ~ E I V •

can be uniquely written

in the form

ow

U.

We have ,u )

is the adjoint o f

is the conjugate o f

For a fixed

(v[i,v,X],u) implies

the

(or a ring or s vector

V, and for each

(2)

(u E U) [i,x,k]

in

U, and

(f E V*) [i,v,X ]

and - -

which

is to take

g.

to be the zero function

that any nonzero vector

in V*. -PROOF.

then

The identity

will denote

a division

IV) and Kothe

of

[i,v,X]*f = fuiV(vkf)

u. i

V*,

6-function

i E I U, ~ ~ 4-, ~ E IV, define

= (v,ui)vv X

li,~,%]*u = uix(v

where

= la E A I a ~ B}.

l X.

be s fixed nonzero vector

(7 6 A-, % ~ IV; similarly

1.2.1 LEMMA.

where

image in

by v[i,~,~]

for some

to

~(V).

let there be given s pair of dual vector

subspacea

and define

A\B

be an index set of all 1-dimensional

V~.

V

it suffices

DENSE RINGS OF LINEAR TRANSFORMATIONS

fixed nonzero vector on

its natural

as a subring of

is a semigroup

elements

set of all l-dimensional

[i,~,~]

S

([7], Chapter

this section IU

ring.

inclusion,

U.

for this section are Baer

I,II,V,IX);

Throughout

~u(V)

S- = ~s E S Is @ Oj.

group of nonzero

([6], Chapters

in

is identified with

to identify

in any division

The general

a

will be denoted by

O, we let

multiplicative

U

(v E V)

For the opposite

of

any sets, we write X

0

= v(a*u)

a*U c U.

is an sdjoint

1.5 gives an easier way For

= v(bu)

(2).

u E U, we have for any

= ((v,ui)xvk,u) Further,

= (v,ui)x(vk,u)

for a fixed

(3) f

is the natural

imase o f

ui v E V, = (v,uiv(vk,u))

f E V*, we have

for any

v E V,

(v[i,~,k])f

= ((v,ui)wv)f

= (v,ui)v(v~f)

(Vfu'1)~(v f) = V(fui~(vkf)) which proves

(3).

The next result

is due to Gluskin

1.2.2 THEOREM.

Every nonzero element of

the form

[i,~,k]

function

exists

Let

~ E IV

into

A-

such that a

can be uniquely written

Since

xa = (xf)v X

is linear,

in

each such

$2,u(V).

0 # a E $2,u(V).

Since

~2,u(V)

i E IU, ~ E A , X { I V , and conversely,

is a nonzero element of

PROOF.

V

for some

14].

the range of

(x E V), where

so is

f

and thus

a

is 1-dimensional,

f

is some function mapping

f E V*.

For any

there

u E U, we

get (x,a*u)

= (xa,u) = ((xf)vk,u)

Since

a ~ $2,u(V),

u E U

such that

such that

we must have

: x(f(vk,u))

a*u { U

(v,u) # O, we see that

xf = (x,ui~)

for some

a =

and all

Choosing

Then there exists x 6 V.

i 6 1U

Consequently

(x E v),

[i,v,k ] .

Suppose (x,uj) @ 0

f(vk,u) E nat U.

f 6 nat U.

~ E h-

xa = (x,ui)vv k = x[i,~,k] so that

so that

(x ~ V).

that

[i,~,X]

so that

There exists 0

(x,ui)~v I = ( x , u j ) 6 v

exists

u E U

such that

i = j.

But then also

The converse

= [j,6,D] # O.

(v ,u) # 0

so that

and thus

x E V

such that

X = ~.

ui~(vk,u ) = uj6(v

Similarly, ,u ) ~ 0

there

and thus

~ = 6.

follows

easily by (1), bilinearity

and nondegeneracy

of the

bilinear form, and (2). By direct calculation we see that multiplication [i,~,h]lj,6,~ ] = for al___!elements of

express

Gluskin

[4].

reference.

set

independent

Ul,U2, .. .,u n

corresponding

is given by (4)

~2,u(V).

1.2.3 SUPERLEMMA*. linearly

$2,u(V)

[i,~(v ,uj)6,~]

The next lemma is of fundamental without

in

Let

set

in

statement

U

importance.

It will be used repeatedly

This lemma can be found in Jacobson

(U,V)

be a pair of dual vector

Vl,V 2 .... ,v n such that

in

V

(vi,uj) = &ij

holds if we interchange

"*Name given to it by some students

there exists for

in class in admiration

spaces.

a linearly

For any independent

i -< i, j ~-- n.

the roles of

U

and

[5], and later in

and

The V.

of its many applications.

PROOF, such that

The proof

is by induction

(Vl,U) = o # 0, so

Vl,V 2,.. .,Vr,Vr+ 1

u I = uo

on -i

be s set of linearly

exist

Ul,U2, ...,u r

u E U

define

in

U

(Vk,~) = (Vk,U)

-

If

n = i, then there exists

independent

such that

r ~ = u - ~ u.(v.,u). i=l i l

n.

has the property

(vi'uj)

vectors

= 6i~i

for

(Vl,Ul) in

V

= i.

u E U

Next

for which

1 _< i, j _< r.

Then for

1 < k < r, we obtain

~ (Vk,Ui)(vi,u) i=l

= (Vk,U) - ( V k , U ) = 0

let

there

For every

(5)

and (Vr+l,U)

r ~ (v +1,u.)(v. u) i=l r i i i'

= (Vr+l,~)

+

= (Vr+l,u)

+ (

¢

"=l

(Vr+l,ui)vi,u).

(6) r

If

-- = 0 (Vr+l,U)

dicting some

linear

u; let

for every

independence

of

tr+ I = uT

It follows

that for

Now suppose and

1 < k,j < r,

(Vr+l,Uk)

Vl,V2,

..,v n

are vectors

in

Hence

(Vr+l,~)

by (5) we have

= (Vr+l,~k) + (Vr+l,Uk)

1 _< i,j _< r + i, (vi,tj)

that

Ul,U2,...,u n

Vl,V2,...,Vr,Vr+ I.

For

= O, and by (6) we obtain

~ (v r+l ,ui)vi Vr+ I = i=l

u ~ U, then (6) implies

= 6ij

where

is a linearly U

for which

independent

if

(vi,u j) = 6ij.

= O.

1 < j < r.

set of vectors

in

If j=l~u.o.j3 = 0,

i < i < n, we have

tablishes

linear

of

which

es-

Ul,U2,...,Un.

The proof of the last statement For

for

= (vk,tr+l)

so that (Vr+l,~k) t.j = ~j

then for

1.2.4 THEOREM.

= ~ # 0

(Vk,Uj)

n n 0 = (v., ~ u ~.) = ~ ( v . , u ) ~ . = o i i j=l J J j=l l 3 J

independence

contra-

of the lemma

is symmetric,

n = 1,2,...,

~n+I,u(V)\~n,u(V)

= {k~=nl[ik,~k,Xk ] lik E IU, ~ E a-, X E I V , nminimal]. n

PROOF. n minimal. We want

First

let

a =

By 2.2, each

to prove

that

with

k~=l[ik,~k,Xk]

[ik,~k,~k]

i k E IU, ~ E & , X k E IV, and

is an element

rank a = n; we will

show that

of

~u(V),

so that

V%l,...,v

n

a E £u(V).

form a basis of

Va. Suppose

that

v%

,...,v%

~iV~l = 6 2 v ~ 2 + ... + 6 ~ V X m of notation).

Then (x~

wh~re

for any

Uil)~iV~l

are linearly

=

all

dependent;

we may assume

~i # O, m ~ n, ~i ~ 0

x E V, (x'uil)~2v~2

+

..

•

+

(x,uil)6mVXm

so that [il,Yl,~l] Consequently

= [ii,62,X2] + ... + [il,~m,l m] .

that

(by a suitable

change

V

i0 n

m

a = k~=2[ik,~k,hk ] + j~=2[il,~j,hj]. For any

x E V

and

2 < j < m,

x([ij,vj,hj] + [il,6j,h j] = (x,uij)Vjvxj + (x,uil)6jvxj = (x,uljvj•. +Uil~j)vh=j = (x,u~j)~jvhj for some

%j E I U

and

~j ~ A, so that m

n

a = k~_2[£k,~k,hk ] = contradicting

the minimality of

A similar proof shows that Further,

Uil

Thus 'Uin

v

,...,v h are linearly independent. hi n are linearly independent.

n x E V, xa = k~=l(X,Uik)~kVhk

for any

subspace generated by i < k < n. -i y = ~k x; then

There exists

x E V

n -1 ya = j~__l(Yk x , u i =

which proves that each

vh

j

such that

)?jvh.

Va.

is in Va

Therefore

proof shows that each Conversely,

let

a

maps

rank a = n

Uik

is in

(x'ui.) = ~jk j

-1

j

= ?k YkVhk

and completes

k a basis of

and thus

V

into the

Vhl,...,v hn

Let Let

n.

+k=m+l~ [ik,Vk,hk]

and thus

a'U, so

a E ~n+I,u(V) \$n,u(V)

=

for

i < j < n.

vh k

the proof that

v

,...v~

is n A similar

hi a E ~n+I,u(V) \~n,u(V).

Uil'''''Uin and let

is a basis of ~iVhl ,

,OnVhn

a*U. bea basis of

Va.

Then v , . ° . , v is also a basis of Va. For any x 6 V we have n hi n xa = ~ (xf,_)v, where fl,...,fn are some functions, fk:V ~ g. By linearity of k=l K Ak a

and linear independence

of

v

,..•,v h , it follows easily that each fk is n linear, so that fk E V*. x E V, u E U, we have n n (x,a*u) = (xa,u) = ( ~ (xf~)v~ ,u) = X(k~=ifk(Vhk,U)) k=l ~ ^k n where a*u E U so that ~ f, (v, ,u) E nat U. There exists u k E U such that k=l ~ ~k n and thus fk (vh ,Uk) = 6jk for i _< j,k <__ n. Hence fk = ~ f.(v h ,Uk) E n a t U j=l j j _ J is the natural image of u ~ for some i k E I U and ~k E A . It follows that hi For any

ik k

n

n

xa = k=l~(x,uik)~kV~k = X(k=l~ Iik,Vk,hkl) not minimal,

then

a

and thus

admits also a representation

a =

n ~ Iik,~k,Xk].

k=l o f t h e form

If

n

m

a = j~l[tj,6j,~j]

is

ii with

m < n

Therefore

m

and

this implies n

minimal.

n

ii)

contradicting

Let

a =

}

rank a = n, n >

m.

then the followin$

are equivalent.

,...,u i }

are both linearly

n

independent.

rank a = n .

In such a case, V X l , . . . , V k n so

form a basis of

Va

and

U.zl,...,Uin

form a basis of

rank a = rank a* = n.

PROOF.

i) = ii).

ii) = iii). Hence

~ [ik,Nk,Xk]; k=l

~Uil

and

n

a'U,

that

is minimal.

(~Xl ..... v x

iii)

the hypothesis

is minimal.

1.2.5 COROLLARY. i)

In the first part of the proof, we have seen that

rank a = m ,

that

This was established

There exist

xka = ~kVkk E V a

xk 6 V

and thus

linear combination

of

that

is a basis of

v

,...,v k

Xl

such that

(Xk,U i ) = 8kp for 1 ~ k,p ~ n. P Since also every element of Va is a

Vkk 6 Va.

Vkl,...,Vkn

and these are linearly Va.

In particular,

independent,

it follows

rank a = n.

n

iii) ~ i).

This follows

The remaining If

in the first part of the proof of 2.4.

~

subring of

statements

is a ring and ~

immediately follow

A

A.

+ a I+ ...+ a --

group closure of

--

--

from the proof of 2.4.

is a subsemigroup

generated by

are of the form

from 2.4.

Since with

A a

n

of

~'~, by

is s semigroup, ~ A, so

C(A)

C(A)

we denote

the elements

is actually

of

the C(A)

the additive

i

A.

1.2.6 COROLLARY. ~u(V) = nU=I~n,u(V) = C(~2,u(V)) an__~d ~u(V)

is a right ideal of

PROOF.

The first equality

and the fact that

-[i,7,k]

opposite

follows

£(V). is obvious.

The second equality

follows

from 2.4

so that with each element, $2,u(V) contains n its negative. If a E ~u(V), then a is of the form a = ~ [ik,Tk,kk] , and (3) k=l implies that for each k, [ik,~k,kk]*V* ~ nat U, so the same is true for a'V*. The inclusion

=

= ~a E ~(V) I a'V* ~ nat U}

[i,-~,k]

from 1.5.

If

(ab)*V* = a*(b*v*) ~ nat U, and hence 1.2.7 DEFINITION. let of

i (A) B

For

be the idealizer

having

A

A of

as an ideal.

a ~ ~u(V)

a subsemigroup A

in

and

b 6 £(V),

(subring)

B, i.e.,

of a semigroup

the largest

for all

a 6 A}.

(ring)

subsemigroup

It is easy to see that in both cases

iB(A ) = [b E B I ba,ab E A

then

ab 6 ~u(V). B,

(subring)

12 1.2.8 PROPOSITION. i)

ig(V)(~2,u(V))

PROOF•

its natural

image in

vk E V

letting

Let

V*.

i~(V)($u(V))

is an ideal of

i) By 1.3, $2,u(V)

gu(V) ~ ig(V)(~2,u(V)).

exists

ii)

= gu(V).

gu(V)

a E i~(V)(~2,u(V)),

It suffices

such that

vku ~ 0.

~ = O(v~u) -I, we obtain

= £u(V).

to show that

Then

which

implies

that

0 # u E U, and identify

u = uio

a*u E U. for some

[i,v,k] E ~2,u(V)

[i,v,k]*u = uiv(vku ) = (uio)(vku)-l(vku)

Since

U

with

u ~ 0, there

i 6 U, ~ E ~ ;

and = uio = u

so that a u = a*([i,v,k]*u ) = (a*[i,~,%]*)u since by hypothesis we have ig(v)(~2,u(V)) ii)

= (a[i,~,k])*u E U

a[i,y,%] E $2,u(V).

Thus

a E ~u(V)

and hence

~ gu(V) •

The proof is an easy modification

of the proof of part i) and is left as

an exercise. 1.2.9 DEFINITION• Then

S

and any set

i < i < n.

2-fold

n

be a positive

is said to be a n-fold

Xl,...,x n for

Let

in

we say doubly

for every positive

1.2.10 PROPOSITION.

For

suppose

kk E I V

n ~

dim V < n, then

dim V ~

yl,...,y n

n

and let

i ~ k < n.

Further,

S

Tk ~ 0

and

i, ~n+I,u(V)

in

There exist

i < k,p < n. _

in

the subspace

a E ~n+I,U(V). ~n+I,u(V)

of

V

Then ui

The next

Let

a =

if it is n-fold

transitive independent

Yk = ~kVk k

E U ~

by default. set and

for some

such that

o k E 4,

(Xk,U i ) = ~kp~k P

[ik,~kl~k,~k];

then

So

Va

for

is contained

k=l

by

v i,

I < k < n, we obtain

~u(V)

This follows

.,v n

so that

rank a < n

and hence

xka = (Xk'Uik)~kl°kv~ k = Yk

of

is a dense rin$ of linear

immediately

theorem "locates"

linear transformation

and instead of

and

transitive.

1.2.11 COROLLARY. PROOF.

set xia = Yi

is n-fold transitive.

is n-fold

be a linearly

V.

_

generated

For

is n-fold

independent such that

is dense

P some

~(V).

[4].

~n+I,u(V)

Xl,...,x n

be any set of vectors and

a E S

we say transitive,

transitive.

is due to Gluskin

If

a subset of

integer n.

The next proposition

PROOF.

S

if for any linearly

V, there exists

Instead of 1-fold transitive,

transitive,

transitive

transitive

yl,.°.,yn

integer and

transformations.

from 2.6 and 2.10

all transitive

rank i; it appears

subrings

to be new.

of

~(V)

containing

a

13 1.2.12 THEOREM. linear

Let

transformation

~

of

be s transitive

rank i.

U = ~ui~ 6 V* I~ E ~, (where

[i,~,~]

t-subspace

of

PROOF. and

is written V*

with

for any

relative

By hypothesis

£(~,V)

contsinin$

U # 0.

for some

~ E ~-, ~ E IvJ

to the dual pair

(V*,&,V))

Su(V) = ~ N$(V) ~ Let

[i,?,~] E ~

there exists

a ~ ~

is the unique

~ £u(V).

with

~ # 0.

such that

For any

(~vx)a = ~v

.

~ E IV Hence

x E V, we have x([i,v,~]a)

so that

= (xui)v(v~s)

[i,~,B ] = [i,~,~]a ~ ~.

[i,~,D] E ~ Let

for any

0.

Hence

= (xui)~v D = x[i,~,~]

Hence

[i,~,~] E ~

with

V # 0

implies

that

~ E &, D 6 I V •

ui~,ujT ~ U

contains have

[i,~,~] 6 ~

the property

~ E ~, by transitivity

subrin$ o f

Then

and

t = u.o+u.~. If t = O, then t ~ U since U J t # 0, so t = Uk~ for some k ~ IV, , V E A-.

suppose

[i,o,~][j,~,~]

~ ~

for every

x([i,~,~] + [j,~,~])

~ E IV .

For any

We

x E V, we obtain

= (xui)ov ~ + (xuj)Tv~ = (x(ui~+uj~))v = (X(UkV))v ~ = x[~,V,~ ]

and thus U

[k,~,h]

is a subspace Let

0 ~ x6 V

[k,6,~] 6 ~ a ~ ~

= [i,o,~] + [j,T,X] ~ ~. of

V*

since

and

y ~ V.

for all

such that

t = Uk~ ~ U

closed under

Further,

There exist letting

z ~ V

y = ~v

k

proves

that

such that

such

and

which

scalar multiplication.

We know that there exists

~ E &, D ~ I V,

xa = z.

But then

it is obviously

that

zu k # 0

and

~2 = ~ ~ 52 V *(V)'

we

obtain x(a[k,(ZUk)-l~,~]) where

a[k,(ZUk)-l~,~]

particular, implies

if

that

E ~2

y ~ 0, there xu i # 0.

= (ZUk)(ZUk)-l?v

by 1.3.

Hence

exists

Since

32

= y,

is s transitive

[i,~,~] E ~2

such that

u. E U, it follows

that

U

semigroup.

In

x[i,v,~ ] = y

which

is a t-subspace

of

V*.

i

From the definition

of

U

and 2.4 it follows

32 = i [i,~,X] E $2,v,(V) Since

~2,u(V)

is an ideal of

by 2.8, part i), we have

~

By 2.6 and (7), we have and hence

~u(V) ~ ~ N ~(V).

gu(V),

l ui E U} = ~2,u(V). we have

~ ig(V)(~2,u(V)) ~u(V)

that

that = gu(V)

= C($2,u(V))

Conversely,

let

~2

(7)

is an ideal of whence

= C(~2) ~ ~

h~.

since

0 # s E ~ n ~(V).

~

is a ring

Then

n

a = k~= [ik,~k,Xk] i V%l ,...,v ~n

and we may suppose

is linearly

independent

that

n

and hence

Hence

~ ~ ~u(V).

is minimal.

By 2.5,

there exist

u j~ E U

the set with

the

14 property

(V%k,Uj%)

-i [jk,Ok ,%k] ~ ~

= 6k~O k

for

for some

1 < k < n

ok ~ 0

and

i < k,~ < n.

Consequently

and thus the element

• -i • -i a[Jk,O k ,Xk] = [ik,~k(V%k,Ujk)O k ,Xk ] = [ik,~k,kk~

is also in

~,

so by (7), we have [ik'~k'Xk]

for

1 < k < n.

But then

~ ~ ~ $2 (V) = ~ 2 = ~2,U (V)

a C C($2,u(V))

= ~u(V)

by 2.6.

Therefore

n a(V) E ~U ( v ) " Let

U~

theorem.

be a t - s u b s p a c e

Then

au(V) = au,(V )

u i o E U, t h e n quently

of

[i,~,k]

U c U~

V*

which a l s o

so

[i,o,k]

and by symmetry a l s o

The equivalence

of parts

E ~2,u,(V)

the rings in the preceding

of the

Hence i f

and t h u s

uio E U'.

U ~ C U, w h i c h p r o v e s u n i q u e n e s s o f

We are now able to characterize ways.

the requirements

a2,U(V) = a2,u,(V).

and t h u s a l s o

E a2,u(V),

satisfies

Conse-

U.

theorem in several

i), ii) and iii) is due to Jacobson

[3], part iv)

is new. 1.2.13 THEOREM.

The followin$

conditions

on a subrin$

~

of

£(A,V)

are

equivalent. i)

~u(V) c ~ c £u(V)

for some t-subspace

ii)

~

is a dense ring containing

iii)

~

is a doubly transitive

iv)

~

is a transitive

PROOF.

i) = ii).

U

a nonzero

ring containing

ring containing

This follows

of

V*•

transformation

of finite rank.

a transformation

a transformation

of

of finite rank.

rank i.

from 2.11 since a ring containing

a dense ring

is itself dense• ii) = iii). Trivially. iii) = iv). Suppose

n>

I

Let

n = min Irank a I 0 # a E ~j.

and let

basis of

Va

By double

transitivity

and let

xib = Yi

for

a E ~ Yn-i

be any vector in

there exists

i < i < n-2.

It suffices

be such that rank a = n.

b E ~

Then for any

xa = ~iXl + ... + ~nXn

V

with

such that

Let

to show that Xl,...,x n

the property

for some

~i E &,

xab = ~l(Xl b) + ... + ~ n _ 2 ( X n _ 2 b) + ~ n _ l ( X n _ l b) + ~ n ( X n b ) = ~IYl + "'' + ~n-2Yn-2 + (~n-i + ~n)Yn-i which

implies

that

rank (ab) ~ n-i Yn-i

we have

ab # O.

iv) = i).

(ab)

and

Since

= Xn-i b = Yn-i ~ 0

But this contradicts This follows

ab E 2.

from 2.12.

minimality

Yn_l a = Xn_ I,

Xnb = Xn_l b = Yn-l'

x E V, we obtain

of the rank of

a.

n = i.

be a

let

15 1.2.14 Exercises. i)

Let

~

be

finite field having

a

space of finite dimension n. both

IV ii)

and Let

IV,

mn_l ~

have

[ik,~k,~k] £ k=l be nonzero elements of ~U(~,V).

a)

(Uil

[l

b)

vXI

= B

and

~i

and

V*

(~,V) have

be a vector

mn

elements,

and

t,6t,~t] , with n and m minimal, t~l[J = Show that a = b if and only if m = n and

n × n matrices

• .. u i n ) = (Ujl

elements and V

elements.

a. =

there exist invertible

m

Show that both

b =

A

and

B

over

&

such that

"'" U j n ) A ,

,

LVXnJ c) ~i ~2

0 B =

A

. . 0

?n

The entries of the row and column matrices above are vectors, with square matrices over

A

and the entries of the resulting matrices iii) ef = 0

A set

idempotents minimal,

E

for any

are again vectors.

of nonzero idempotents of a ring e,f E E, e ~ f.

~

is said to be orthogonal if

Show that the sum of a finite number of orthogonal

is again an idempotent.

£ [ik,~k,)~ k] E ~u(V) with n k=l find necessary and sufficient conditions in order that a be an idempotent.

Show that for any

n >

there exists a set e = el+e2+...+e iv)

the m u l t i p l i c a t i o n

is performed according to the usual row by column rule

0, e E ~u(V)

el,...,e n

For

a =

is an idempotent of

rank n

of o r t h o g o n a l idempotents in

if and only if

~2,u(V)

such that

n

Prove that every nonzero ideal of a transitive subsemigroup of

g(V)

is

itself transitive. v)

Prove that every nonzero ideal of a dense subring of

~(V)

is itself

dense. vi) Va

Show that an element

s

of

~(V)

is idempotent if and only if

elementwise fixed. vii)

For

a,b E g(V),

prove that

frank a - r a n k b l <-- rank (s + b ) .

a

leaves

16 viii)

Let

(U,~,V)

be a dual pair, I

be a nonempty subset of

IU

and

S = {nit I i E I, T E A], T = [ li,~,~] ~ 3u(V ) I i E I]. Show that ix) a ring

S

is a subspace of

U

if and only if

The least positive integer

n

C(T) n 3u(V) = T.

for which

nr = 0

~, if such exists, is the characteristic of

have characteristic zero.

Show that

g

and

for all elements

~; otherwise

3u(A,V )

~

r

of

is said to

have the same characteristic.

For further results on dense rings of linear transformations see Dieudonn~

[2],

[3], Gluskin [4], Jacobson [3], [4],([6], Chapter IX, Section 10),([7], Chapter IV, Section 15), Ribenboim ([i], Chapter III, Section 2). found in Erdos

1.3

ONE-SIDED IDEALS OF

We fix a pair of dual vector spaces V~

Some related material can be

[i], Rosenberg [i], Rosenberg and Zelinsky [i], Zelinsky

a subspace of

[3].

3u(V)

(U,A,V), and for

UI

a subspace of

U,

V, we use the notation = ~a E 3u(V ) IVa c- V ' ,

~,(V')

a*U ~ U'}.

It is clear that

,%u,(v') = 1%,(v) n gu(v'). We will sometimes write Clearly

~u,(V)

also of

3.

3

instead of

is a right ideal and

3u(V ) ~U (V ~)

(1) and

32

instead of

32,u(V ).

is a left ideal of

We will consider only right ideals and subspaces of

$1u(V) U.

lation and proof of dual statements for left ideals and subspaces of left as an exercise; for this two different proofs are possible:

and thus

The formuV

will be

direct by re-

taining the notation already used, or using 1.2 one may consider the isomorphic copy of

~

in

£(U)

1.3.1 LEMMA.

and then dualize the proofs to be presented here.

We have n

~,(V') PROOF.

Let

= [k~=l[ik,Wk,Xk ] =

By symmetry and using (i), it suffices to show that n ~ , ( V ) = ~k~__l[ik,Vk,Xk ] lUik E U ~, ~k E A, vXk E VJ.

0 # a E ~u,(V).

Then

a*U c U I --

so that that

l Uik ~ U j, ~ E A, VXk E V'}.

and

a =

i

[ik,~k,k k]

where

(2) n

is minimal

k=l

v

,...,v X are linearly independent by 2.5. There exist u. E U such kk n Jp (Vkk,Ujp) = 6kpO k for some ~k ~ 0 and i <_ k,p _< n. Hence for I ___ k _< n,

we have -i [ik,Vk,% k] = a[jk,o k ,X k] ~ ~ , ( V ) since

~UI(V)

is s right ideal of

~.

But then

[ik,~k,~k]*U ~ U ~

which implies

17 that u. ~ U ~ and thus ~k inclusion is obvious.

a

is in the right hand side of

(2).

The opposite

The next result is new. 1.3.2 PROPOSITIONthen

form,

(Ut,~,V ~)

Suppose that

form to

(UI,g,V t)

V~X U ~

From the definition of in

UI

alv,

If

~ l ( V ~)

is equal to

b ~ ~u~(V~),

sentation of

a*Iul,

b, we extend

b

(x E V). (u E U)

and

~

maps

have

a =

suppose 'that

~ k=l [ik'~k'~k ]

that

Uil

, ...

n

c

a~ # O.

Consequently

Fixing this repreV

by defining

defined by

the adjoint of

a

in

0 # a E ~ut(V~).

(U~,V ~)

for

U.

Consequently

Then by 3.1, we

1 < k < n, and we may

is a dual pair, and by 2.5 we have there exists

v E V~

such that

Hence

n va = ~ (v,u i )~kVx = k =I k k (v'uil)~iV~l and

is a homomorphism.

on all of

is zero, let

Since

i < k < n.

by

~uI(VI ).

are linearly independent

for

a

u. E U j, v E V~ ik Xk

is minimal.

,Uin

(v'ui k) = ~kl

with

~

~

n b = k~_l[ik,?k,Xk ] . =

Then the function

onto

To show that the kernel of

~

of the

a~ E ~ul(V~), where the adjoint of

it follows that

and it is clear that

is evidently

~ I ( V I)

the restriction

(a E ~ u , ( V ' ) ) .

to a function

n cu = k~=lUik~k(V~k,U )

i.e.,

Define the function

then by 2.6 we have

xa = k=l~(x,uik)~kV k

a~ = b

is a dual pair,

is nondegenerate.

~:a --

a~

is a dual pair with the induced bilinear

~uI(V I) ~ ~ut(VI).

PROOF. bilinear

If

the kernel of

~

~ 0

is zero and

~

is an isomorphism.

The next lemma will be quite useful. 1.3.3 LEMMA. PROOF. for any

If

i E IU

For any

a ~ £u(V),

a # 0, then

xa @ 0

and

Since for every

(xa,uj) ~ O.

for some

= (Y,Ui)?(vxa)

i E IU, there exists that

so that

y E V

[i,~,X]a ~ O.

Hence for any

= (xa,uj)~vx # 0

~2 a = 0

or

a~ 2 = O, then

0 # x E V.

Hence

a = 0.

x = ~v X

and thus

y E V, we obtain

y([i,~,X]a)

xa # O, it follows

if

= (Y,Ui)(xa). such that

Further,

~ E ~-, ~ 6 IV, we obtain a[j,~,X]

# O.

(Y,Ui) # 0

there exists

j E IU

x(a[j,~,X])

and we have such that

18 Recall

that a lattice

one-to-one

correspondence

isomorphism

of a lattice

which preserves

preserves

either meet or join or the partial

lattices,

consult

Sz~sz

1.3.4 THEOREM.

[i].

a l l risht

U'

and

isomorphism

(U I

is

of the lattice

B

is a

(it suffices

For matters

is due to Jacobson

that it

concerning [5].

subspace

a

o__ff U)

of all subspaces

of

U

onto the lattice

of

3u(V ).

We have already

Un

onto a lattice

The function

ideals of

PROOF.

order).

The next result

)<:U I ~ % l ( V ) is a lattice

A

both meet and join

are distinct

observed

subspaces

Hence

for any

that

[i,~,~] E % , ( V ) \ % s , ( V ) .

that

of

U'

[i,~,~]*U ~ U'

Consequently

and

UH

is a right

U, we may suppose

v k E V, ~ E A-, we have

For any subspaees

%,(V)

of

~<

ideal of

3.

If

that

u.l E U ' \ U '~. [i,7,k]*U ~_ U n, so

and

is one-to-one.

V, we have

% , (v) n ~,, (v) = 3u,n u" (v) which

implies

suffices form

that

)~ is a meet homomorphism.

to show that

%,(V) Let

){

for some

31

is onto,

U'.

be a nonzero

Note

right

i.e., that

If

0 # a E ~, ~i~.

then

ideal of

a3 2 ~_ ~ N 3 2

By 3.3, we also have

zero right

ideal of

32.

[i,6,p] Hence

[i,?,k] E ~I2

p ~ I V,

u.j E U

since

= [i,v,k ] [j,

with

V # 0

As in the proof of 2.12, ~2 = ~

for some '~

that

such

the proof,

ideal of

3

it

is of the

$, and let ,{ ~ A-, k ~ IV].

is a right

a32 # 0, which

It follows

P E IV, then there exists

to complete

0 = ~o(V).

U' = [ui(~ ~ U Io E A , [i,~{,k] ~ 31

of

Hence

that every right

implies

U' # 0.

that

ideal

that

If

and

32

is an ideal

312 = ~ ~ 32

is a non-

[i,~,k] 6 312' 6 E A,

(v ,uj) = o # 0, so that

o-1 -1 6 V ,p] 6 ~2"

implies

that

it follows

N 3 2 = ~ [i,v,k]

that

[i,6,p] E ~2 U~

for any

is a snbspace

of

6 ~ fl, U

and

l ui E U ~} n

%,(V) = C('~2)

so that minimal.

There exist

i <__ k,p _< n.

c_~. u.JP 6 U

t~

Therefore

0

# a = k~__l[ik,Vk,kk ] _

such that

(Vkk,Ujp)

be in

= ~kp(~k

31

with

for some

n dk # 0

and

Hence [ik,Vk,kk]

since

Let

is a right ideal. 31 = % 1 ( V ) .

= a[Jk,okl~k,kk ] ~ Thus

u. ~ U ~ ik

which by 3.1 implies

that

a E "~u'(V)"

19 1.3.5 DEFINITION.

For

~

a ring,

let

~2 = { i a.b. I ai,bi ~ ~i}. i=l z l It is clear that a,b E ~t.

j~2

is an ideal,

and that

A ring

~ a

is called simple of a semigroup

if

0

is said to be regular

(or a ring)

S

is regular

regular.

This concept was introduced

1.3.6 LEMMA. PROOF.

A semigroup

Let

.,Uin } v

for some

(regular)

by von Neuman

~VXl,...,VXn}

and

are linearly

I

n

a = aba are

rings are also

minimal.

independent

n

Then both

by 2.5

E V and u. E U such that (v , u ) = ~kpOk, ~k ]p ~k ip o k ~ O, ~k ~ O, and 1 < k,p < n. Hence

a

if

if all its elements

[i]; regular

is a regular simple ring. n 0 @ a k ~ l [ i k ' ~ k ' % k ] ~ ~u(V) with

n

and thus

~ 2 ~ O.

rings.

There exist

(v k,Ujp ) = 6kpG k

n

(k~=l[ik,Yk,kk])(p~=l[Jp,';l~;l°;1 If

for all

gu(V)

, . .

nonzero

ab = 0

are its only ideals and S

b E S.

known as von Neumann

and

(or a ring)

for some

vectors

if and only if

Such a ring is called a zero ring.

An element

~Uil

,~2 = 0

n

Dp])(k~=l[ik,~k,kk ]) = k~__l[ik,~k,kk ]

is regular. is a nonzero

ideal of

gu(V),

then by 3.4 we have

subspace

Ul

of

U, and by the dual of 3°4, also

nonzero

subspace

Vl

of

V.

V~ = V

and thus

I = ~u(V).

Consequently,

I = ~I(V)

Since

is regular,

~u(V)

1 = ~uI(V)

I = ~u(V I)

= ~ ( V ~)

for some

for some

so that

U ~ = U,

it is not a zero ring,

and

hence must be simple. 1.3.7 DEFINITION.

~£(~)

For

any ring,

I

the set

= 0

for all

a E ~j

~r(~) = [r E ~ I ar = 0

for all

a ~ ~]

=

~r E ~

is the left annihilator

is the right annihilator

of

of

It is clear that both appears

~

to be new.

ra

~,

~. ~(~)

and

~r(~)

are ideals of

~.

The next result

20 The followin$ statements

1.3.8 THEOREM,

concerning ~ subspace

U~

of

U

are equivalent.

i)

(U~,V)

is a dual pair.

ii)

~u~(V)

is a simple ring.

iii)

~%(~,(V))

= 0.

iv)

~,(V)

is transitive.

v)

~,(V)

is a regular ring.

If any of these conditions holds, PROOF. ~I(V)

i) = ii).

~ ~uI(V).

that

~(V)

Since

is simple. By simplicity of

iii) = iv).

Let

x,y E V

is the zero function.

that

For any

[i,o,X]s ~ 0

for some

~uI(V), we must have

with

So suppose

i E IUI , we have

[j,~,~] E ~uJ(V).

a ~ ~uI(V).

iv) = v).

Since

E ~u~(V). ~uI(V)

transformation of rank i. such that ~u~(V)

~(V)

= ~u~(V)

regularity,

Let

Hence

~uI(V )

a E ~uI(V)

that

is a dual p a i r

~e consider next principal 1.3.9 THEOREM.

[i,o,~] [j,~,~] ~ 0

(v ,uj) # 0.

~u~(V)

For any

0

o # 0,

implies for some

Thus

is transitive.

since all elements of

and thus there exists

where

where

and the hypothesis

is a nonzero right ideal,

[i,~,~]a # 0 (U~,V)

xO = y

x = ovk, y = ~v

and

= 0.

= ~(v x,uj)(v x,uj)-IO-I~v~ = y

u.l E U r ' ~ E ~

there exists

y = 0, then

But then

j E IU~

~%(~l(V))

it contains a linear

Thus by 2.12, there exists a t-subspace

is regular by 3.6, so

v) = i).

If

Hence

[i,o,l] E ~u~(V)

It follows that

[j,(v~,uj)o-l~,D]

x # 0.

y ~ 0.

x[j'(vX'u')-Io-I~'~]j where

~u~(V) ~ ~u~(V ). is a dual pair, 3.2 implies that

This proves the last statement of the theorem and by 3.6 implies

ii) = iii).

# 0.

then

(UI,V)

~(V)

U~

of

V*

are of finite rank.

is regular. v~ ~ V.

Then

such that u.j 6 U ~

[i,~,~] E ~ ( V ) ,

and by

[i,~,~] = [i,~,~]a[i,~,~].

such that

Hence

(v%,uj) ~ 0, which implies

ideals.

s E 5, th___~e~rincipal

risht ideal senerated b_~y a

equals a~ = ~a,u(V). PROOF. with

If

n minimal.

a = 0, the assertion

But

is trivisl,

Then s typical element of

aS

so let

0 ~ a =

is of the form

~ [ik,~k,Xk] k=l

21 m

n

m

at~=l[Jt,~t,P t] = (k~='l[ik,~k,Xk])(~l[Jt,6t,Pt ])

(3)

= k= ~ I t~=ml[ik,~k(VXk,Ujt ) ~t'~t ] with

Jt E IU, ~t

E ~, Pt ~ IV"

P q=~l[%q'°q'~q]

is of the form U~q = a*(Usq~q)

E a*Uo Hence for q Usq E U, Tq 6 A so that

for some n

U~q

On the other hand, a typical element of

with

u~

.

*

(a*u sq)~q = k~=l[lk,~k,~k ] _

Consequently,

for any

%a*u (v)

I < q < p, we have

n u sq~q = k=l~U.lk~k(V~&k~USq)~q.

x E V, we have

n P P P x ( ~ [%q,Oq,~q]) = ~ ( x , u ) 6 v = ~ ~ (x Ulk)Vk(Vkk,USq)TqOqV q=l q=l ~q q Vq q=l k=l ' " p n = x(~ ~ [ik,Vk(VXk,U )~ O ,~q]) q=l k=l Sq q q

q

so that P q~=l[~q,Oq,~q] E IU, "rq E ~ - , q a~ = ~a.u(V).

with

s

The set

s~

P = k=l ~ ~ [ik ,Vk(VXk, u s q )~ qOq, Vq] q=l "0q E I V .

A c o m p a r i s o n o f (3) and (4) q u i c k l y shows t h a t

is indeed the principal right ideal generated by

is clearly a right ideal and

a E aS

since

The next corollary is a supplement 1.3.10 COROLLARY.

U

$

a

since

a~

is regular by 3.6.

to 3.4.

The isomorphism

finite dimensional subspaces of of

(4)

x:U' ~ ~u~(V)

maps .the lattice of all

onto the lattice of all principal risht ideals

~u(V). PROOF.

Let

U'

be an n-dimensional

has a basis of the form for some a*U ~ U ~.

ok # 0

and

Uil , "''Uin

subspace of

Let

i < k,p ! n, and let

On the other hand, if

V~k E V

U

where

be such that

n . -i a = k~__l[ik,~k ,hk ].

u E U ~, then

n u = k~=lUik~k

n -1 n a*u = k=l ~ u.ik o k (v~k ,u.ik ) ~K = k~=lUikVk = u.

n > 0.

Then

U'

(V~k,Uip) = ~kpOk It is clear that

and thus

22

Hence

U' c a*U

and thus

u'x Conversely, a E ~.

if

~u,(v)

=

~

a*U = U'.

1.3.11 REMARK.

~a.u(V)

=

is a principal

Thus by 3.9, we obtain

correspond minimal

By 3.4 and 3.9, we have =

a~.

right ideal of

(a*U)x = ~a,u(V)

It follows

5,

then

~ = a~

from 3.10 that to l-dimensional

risht ideals,

for some

= ~.

and from its proof,

subspaces

of

U

that they are all the sets

of the form R i = {[i,~,X] Similarly

all minimal

IV ~ 4, X ~ Iv3

left ideals are ~iven by

Lk = [ [ i ' v , k ] Further,

L k = ~[i,o-l,k]

(k ~ IV). where

and

(vk,ui) = o # 0

is sn idempotent.

1.3.12 PROPOSITION. has no proper PROOF. vectors

I i ~ IU, ~ ~ ~j

R i = [i,o-l,k]~,

[i,o-l,k]

(i E IU).

V

is a finite dimensional

vector

space,

then

V*

t-subspaces. Let

in

If

dim V = n, let

V, and let

U

be a linearly

Xl,...,x n

be a t-subspace

of

V*.

independent

set of

There exist

f. E U J independent.

such that

x.f. = 6.. for i < i,j < n and fl'''" f are linearly Hence i j lj --' n dim U ~ n and since U is a subspace of V* and dim V* = n, it follows that U = V*. 1.3.13 COROLLARY. finite dimension

If

(U,V)

is a dual pair and either

U

or

V

is of

n, then the other is also and each can be identified with the dual

of the other. For n xn

~

any ring and

matrices

over

~;

1.3.14 COROLLARY. satisfies

d.c.c,

n

a positive

integer,

also write d.c.c, If

~

denote the ring of all n instead of "descending chain condition".

is a subrin$ of

PROOF.

Let

~

~U(&,V)

for either left or risht ideals,

the same finite dimension, ~ = ~(~,V) ~ ~ d i m V ' and risht ideals,

by

~

containin$

then both

satisfies

U

d.c.c,

~U(~,V) and

V

and

are of

for both

left

and all its left and risht ideals are principal. U1 ~ U2 ~

...

be a descending

chain of subspaces

of

U.

Then

by 3.4 we have that of

~ i (V) ~ ~U (V) ~ ... is s descending chain of right ideals 2 Since each ~ui(V ) is also a right ideal of ~ and ~ satisfies

Su(V).

d.c.c,

for right ideals,

there exists

i > n

and thus by 3.4,

U. = U

-either

1

U

or

dimensional,

V

satisfies

n

for all n

d.c.c,

such that i > n.

~ui(V ) = ~ n ( V ) Consequently

for all

by 3.4, or its dual,

-

V

is finite

and hence by 3.13 both are of the same finite dimension,

for subspaces,

so either

U

or

and we may

23 write

V* = U.

Using 1.4 we obtain

Again by 3.4 and its d u a l

~

£u(V) = ~(V)

satisfies d.c.c,

and hence

~ = ~(V) = ~(V).

for both left and right ideals.

The

last statement follows from 3.10. 1.3.15 COROLLARY. risht ideals; V

(and thus also PROOF.

~u(V)

satisfies d.c.c, for principal

it satisfies d.c.c, for risht ideals

follows from 3.10 since a vector space evidently

for finite dimensional subspaces.

statement follows from 3.14, and the converse Here,

as before,

PROOF.

U.

Let

such that On

A

Fo___rr a,b E ~(U), A

be a basis of

b~ = u

define

a

if

function

d

aU ~ bU,

aU.

(possible since

The direct part of the second

from 3.10 and the first statement.

the "left" part of the statemenls

1.3.16 LEMMA.

U

if and only if

U) is finite dimensional.

The first statement

satisfies d.c.c,

left and principal

(or left ideals)

follows from dual statements. then

For every

u 6 A

A c aU c bU).

by letting

du = ~,

and extend d to a linear transformation on U. n ax = k =~lukOk= for some u k E A, o k E g, and thus n

a = bc

for some

choose an element

Extend

A

let

map

d

For any

n

c E ~(U).

to a basis B\A

~ B

of of

onto zero,

x E U, we have

n

= bd(~k=lUkOk) = b[~l(dUk)°k]= = b(~l~k°k)=

bd(ax)

n

Letting

c = da, we get

For

S

a semigroup

a = bc

with

(or ring)

c ~ ~(U).

let

ES

be the set of all idempotents of

S

partially ordered by e < f Let V'

~(U,V)

if and only if

be the set of all dual pairs

is a subspace of

including the pair

V, dim V ~

If

(A,~) onto

a ! a~

(U~,V~), where

is finite,

under

U~

is a subspace of

U,

the induced bilinear form,

(O,0), partially ordered by

(U~, V~) ! (U#, V~j)

(A,!)

e = ef = fe.

and

(B,~)

(B,!)

if and only if

are partially ordered sets,

is a one-to-one

if and only if

function

~

a~ ! a ~ "

The next result is new. 1.3.17 THEOREM.

The function

~ : e -- (e*U,Ve) is an order isomorphism of

U ~ ~ U ~, V ~ ~ V'.

E~u(V )

~

defined by (e E ESu(V ))

onto

~(U,V).

mapping

then an order isomorphism of A

onto

B

such that

24

PROOF.

Let

k=l There exist

[ik,Wk,hk]

xk ~ V

-i = 6kp~k .

(Vkk,Uip) u

. i I ' "''Uin Let

V'.

=

2

There exist

Since by 2.5,

be an element u i P E U' Uil

'Uin

3.13 also

: n

and hence

Letting dently

dimnU'

e = k=l~ [ik,~[l,kk],k Ve ~ V'.

If

it follows

of

~(U,V)

the set

so that

equation

Similarly

e,f E E~

e*U = U ~

and assume

Hence implies

Let

c 6 ~(U),

lent to:

e~U ~

which

~k ~ 0

dim V' = n, by

Since

forms a basis of

since

(Vkk,Uip)

for some

and

Uj

= 6kpTk.

ok 6 A

Evi-

and thus

kk

e = ef

e~p = f~. and

f'U, Ve ~ Vf

1.3.18 PROPOSITION.

fe

the first gives

means

that

that

to:

e*U = f*U

a ~ £(V) = f'e*.

3.2,

The last

e = f. which can be

e* = f*c

the last statement

e = df is equiva-

e~e ~ &rank e"

3.13,

and 1.4 we obtain

= ~e,u(Ve) = £(Ve) ~ ~rank e"

which

and

eq0 < f~p.

(i), 3.17,

= ~(Ve),(Ve)

and

e = ef = fe

0 # e E E~, we have

= ~e,u(V)~u(Ve)

~e*u(Ve)

Ve = Vf for some

is in turn equivalent

which means

For ~

e~ = (U~,VJ).

e* = f*b = f*(f*b)

By 3.16 and its dual,

Using 3.9 and its dual, e~e = e $ ~ e

Then

e* = f*b

and

e < f

which

d E £(V).

and therefore

together with

The relation

e* = f'e*

for some

PROOF.

that

f = ae

f = ae = (ae) e = e = fe

e,f E E~°

written as

be a basis of

n -i n n ~ (x,u i )~k v = ~ (~ ~ v ,Uik)~kl = ~ OkV = x k=l k ~k k=l p=l P kp Vkk k=l kk

by 3.16 and its dual yields b 6 £(U).

,Uin

and

is a dual pair.

for some

independent

e2 = e

Ve

v kl ,. ..,Vkn

= 6kpT k

n ~ okv

x =

so we must have

(e*U,Ve)

and let

k=l

Ve = V ~.

Let

that

Uil

we see that then

so that

form a basis of

(Vkk,Uip)

--

xe =

Then

1 < k,p ! n

independent

Vkl,...,Vkn

are linearly

x 6 V',

minimal

= p~__l~k(V>k,Ulp)~pVkp

such that

i < k,p < n, and

for

are linearly

e'U,

n

~n

xke = xke

form a basis of

(U',V')

(Xk,U i ) = 6k p P

v 1 ,...,v I n

the vectors

with

2 = k=l ~ p=l i [ik,~k
= e = e

such that

~kVkk By 2.5,

n ~ [i,,~k,~k] ~ E~ k= 1 K

0 # e =

25 1.3.19 DEFINITION. every finite subset isomorphic to

~

F

Uj

~

is a locally matrix rin$ over a ring

there exists a subring

n

Su(A,V)

be the subspace of

U

e E 3

of the proof of 3.10 is an idempotent in

quently

a i E ~U j(V)

aa i = a(abi) = ab i = a i n ( O Vs.) U Vs. i= I l

generated by

an idempotent Letting

b E 3

e = a+b

~

3

and

a

Un a~fU. i=l l

U'

Then

is

constructed in the first part

a i = sb i

1 < i < n.

al,s2,...,s n

a I, ...,a n E eSe.

a3 = ~ , ( V ) .

for some

Next let

V~

b.l E 3.

It Conse-

be the subspace of

By a dual argument, we conclude that there exists

such that

3b = ~u(V ~ '), a i = aib for i < i < n 2 - bs, we obtain e = e, ea = a, be = s, and hence

a i = aaib = (ea)ai(be) 6 e~e

for

([7], Chapter IX, S e c t i o n 15, Theorem 3).

theorem can be found in Faith and Utumi

and

s = ab

1 < i < n.

The above theorem in c o n j u n c t i o n with 11.2.8 implies Litoff's Jacobson

F

A.

with the property

and hence for

if for

containing

such that

generated by the set

finite dimensional and the linear transformation

V

of

to show that for every finite set

3, there exists an idempotent

follows from 3.9 that

~

~

n.

is a locally matrix rin$ over

By 3.18, it suffices

of elements of Let

of

~

for some positive integer

1.3.20 THEOREM. PROOF.

A ring

theorem,

see

Different proofs of Litoff's

[i] and Steinfeld

[i]. We will prove a

g e n e r a l i z a t i o n in 11.2.19 due to Hotzel. 1.3.21EXERCISES. i) ii)

iii)

Find all zero rings without proper ideals. Show that for any semigroup

(or ring)

S,

the following are equivalent.

a)

S

b)

every principal right ideal is generated by idempotent,

c)

every principsl

be a ring and

~

be s right ideal of

~

~

is a right ideal of

~.

v) of

~

left ideal is generated by an idempotent.

Show that every right ideal of iv)

Let

is regular,

Prove that

g(V)

which is a regular ring.

is s regular semigroup.

Show that dense subrings of

3u(V )

are precisely the simple right ideals

3u(V ) . vi)

Let

S

be a semigroup and

e

be an idempotent of

S.

Show that

G e = {x E S Ix = ex = xe, e E xS n S x ] is a subgroup of Ge

S

containing every subgroup of

is called a maximal subsroup of

S.

S

having

e

ss its identity;

26 vii)

Let

e E E£(V).

Prove

that

1 - e

is an idempotent, N e = V ( I - e),

V = N e G V¢, G e = {a E £(V) INs = Ne, Va = Ve} having

e

as its identity and

transformations on viii)

For a)

ix) if = xf

b)

(a*U,Va)

c)

V = N a ~ Va.

linear

(f E V*).

~u(V),

is a dual pair,

(&,V)

and

x E V, let

Prove that the m a p p i n g

~:x ~ x

is called the natural isomorphism of

x)

g(V)

show that the following are equivalent.

is contained in a subgroup of

For any vector space

V**;

is the maximal subgroup of

is isomorphic to the group of invertible

Ve.

a E ~u(V), a

G

V

x:V* ~ g

be defined by

is an isomorphism of

into

V

into

V**.

Prove that the following statements concerning a vector space

V

are

equivalent.

(Hint:

a)

V

b)

V*

is finite dimensional.

c)

V*

has no proper t-subspaces.

d)

~:x - i

e)

~(V)

If

xi)

is finite dimensional.

V

maps

V

onto

V**.

is a simple ring.

is infinite dimensional,

Let

show that c), d), e) fail.)

be a d i v i s i o n ring and

n

be a positive integer•

For the ring

An

a)

find all left (right) ideals,

b)

w h i c h of them are minimal, maximal,

c)

show that

£

is a simple,

n for both left and right ideals.

principal,

simple?

regular ring satisfying d.c.c,

and a.c.c.

Use the results in the text but also give a direct proof w h e n e v e r you can. xii)

Let

e

f = e+g

if

xiii)

If

and

f

for some idempotent •

g

in

is a dense subring of

finite rank, prove that ideal of

be idempotents in a ring

~ ~ $(V)

~

£(V)

~.

orthogonal

to

e < f

if and only

e•

containing a nonzero transformation of

is an ideal of

~

contained in every nonzero

~.

For more i n f o r m a t i o n on this subject see Behrens •

Show that

i

Dzeudonne [2], G l u s k i n Chapter IV, Sections

[4], J a c o b s o n

([i], Chapter II, S e c t i o n 3),

[5],([6], Chapter VIII, Sections 3,4),([7],

15,16), R i b e n b o i m

([i], Chapter III, Sections

2,3), Rosenberg

[i].

27

1.4

IDEALS OF

1.4.1 DEFINITION. ideal if

g(V)

A nonempty

x E S, y E I

implies

AND PRINCIPAL FACTORS OF subset

xy E

I

I

of a semigroup

(yx E I).

If

I

is an ideal of

S

S\I,

~nd

with

I @ S, then the set

if

xy~

I

if

xyE

I

I

is a left (risht) is an ideal of

S

if

S.)

the multiplication:

=fxy

S

(Hence

and only if it is both a left and s right ideal of

is not an element of

[~u(V)

Q = (S\I)

for

(J 0, where

0

x,y E S \ I ,

x*y

x*0 is a semigroup denoted by Let S

called

= 0-0

the Rees Nuotient

= 0 of

S

relative

to the ideal

I

and is

S/I. J(a)

(i.e.,

= 0*x

denote

the principal

the smallest

ideal of

S

ideal generated containing

by an element

a).

a

of a semigroup

It is easy to see that

J(a) = a U a S U S a U SaS, and if

S

Let

has an identity,

J(a) = SaS.

I(a) = {b E J(a) I J(b) # J(a)j.

empty or is an ideal of as an ideal of I(a)

then

= ~,

let

For

C

J(a),

S.

One verifies

The Rees quotient

is called the principsl

J(a)/I(a)

easily that

J(a)/l(s), factor of

where

S

l(a)

l(a)

containing

is either

is regarded a

(if

= J(a)).

+ than

a cardinal number,

let

¢, and for a c-dimensional

C

denote

vector

space

the least cardinal number (g,V),

greater

let

(a,V) = ~b E g(~,V) I r a n k b < ~} together with

the multiplication inherited from g(~,V), where Q is a cardinal + 0 < ~ < ¢ . As usual, we will omit & if there is no need for stressing

number,

the division

ring.

The next result appears

1.4.2 THEOREM. --°f g(V)

PROOF.

g (V)

since

generated g(V)

gb+(v) •

vector

is s principal

That

find principal J(b)

The set {gQ(V) I0 < Q < ¢+j

for a =-dimensional

addition,

g (V)

ideals. by

to be new.

b

space

b 6 g(V)

consists

has an identity.

g(V)

and let

of elements

the set of all idesls

g~(V) ~ gb(V)

ideal if and only i f

is sn ideal of Let

constitutes

V, and

follows

~

from 1.3.

rank b = b.

of the form

__if Q # b.

__In

is not a limit cardinal.

cbd

Again by 1.3, such an element

We will

first

The principal

ideal

where

c,d E g(V)

is contsined

in

28

Conversely, extend Vb by

A

let

to a basis

such

that

(possible

For each

a function

on

c

B\A, of

for any

d

8

choose

C\Aa,

x E V, we have

xa =

n

(xa)dbc Consequently,

° (

i(Yid))bc

a = adbc E J(b)

~ E V

xd = x8

for each

c

and extend ~ aiy i i=l

such

that

on

g(V),

and

into

~b = x

V.

it to a linear

d

Extend if

A8

x E AS,

transformation

~. E A, Yi E A, and I

n

J(b)

and

A

x E A, define

on

thus

n

° ° thus

Va

rank a ~ rank b

xc = x8 -I

for some

of

mapping

since

transformation

by letting

°

and

function

(possible

n

1

be a basis

an element

function

the set

A

be a o n e - t o - o n e

it to a linear

a

Let

independent

by letting

Define on

rank a < b-

Let

and extend

V

arbitrarily

Then

V.

x E AS,

define

C

then

is linearly

A8 ~ Vb).

to a basis

V.

of

A8

since

arbitrarily

define

B

the set

hypothesis).

Now

a E gb+(V),

i°l °iY

= g +(V). b

Next larger

let

than

for some

J

rank b

is a limit

assertions

Let from

But

~(V).

Jacobson

and

then

denote

The next

constitutes PROOF. C ( ~ 2 v,(V)) ,

be the least

with

proves

J = g

Q

(V).

g

then

number + b = Q

by the above.

c E J

that

cardinal

cardinal,

rank b = b

there exists

which that

the semigroup

g

can be

such

If

that

(V) c_ j.

Since

the

The proof of the r e m a i n i n g

(V)

found

provided in Baer

9),([7], Chapter vector

with

the a d d i t i o n

([i], Chapter

IV, S e c t i o n space

V,

inherited

V, A p p e n d i x

i),

17).

the set

+ or

~ <

c

the set of all ideals By 4.2 and

So by 4.2, £(V)

b

For a c - d i m e n s i o n a l

I~ = ¢

= ~(V),

Q

is left as an exercise.

+ ~£Q(V)

let

is not a limit

then

J

we have

IX, S e c t i o n

1.4.3 COROLLARY.

a

for any

rank b < a,

corollary

([6], Chapter

If

b ~ J(c) c

is trivial,

of the theorem

£a(V)

b E J.

J = J(b)

cardinal

inclusion

ideal of

for all

b, and hence

rank b < rank c. opposite

be any

no

1.3, g

(V)

has no other

and of

~

is an infinite

cardinal]

£(V).

these

are ideals of £(V). Since by 2.6, + for ~ ~ ¢ and Q finite is closed under

ideals

than

those

listed

in the statement

addition.

of the

corollary.

We have

seen in 2.2 that

the nonzero

written

in the form

[i,~,X]

where

written

in the form

[i,0,k]

for any

elements

of

This

is a special

case of

=

[i,~(vk,uj)6,~].

the following

can be uniquely

I V , the zero

function

i E IU, X E IV, and the m u l t i p l i c a t i o n

given by [i,~,k] [j,~,~]

$2,u(V)

i E IU, ~ E ~ , h E

construction.

is is

29 1.4.4 DEFINITION. element not in with

Phi

exists that

Let

the value at

% E A

I

G, and let

By

denote

(i,0,~)

We will prove next that for all finite

group

Ln(&)

of

positive

~ ~ A

Then

Let

i E I

such

is a semigroup

is isomorphic

a division

ring and

matrices

over

n

as a matrix

and

to a Rees matrix

&

a positive

integer,

(under multiplication)

linear group of degree

n

over

the is

g.

is new. (&,V)

be a vector space , U

n < dim V.

! t-subspace o f

is an n-dimensional

subspace of

B

B

Vl, ... ,v n

fi__~x! basis

V*, n

Let

n U = [A I A

A.

i ~ I, there

can be considered

subspace

of __

be an

with multiplication

S = ~°(I,G,A;P)

P = (pxi)

is an n-dimensional

B E nv

0

P:A × I ~ G °,

there exists

nV = {BIB

A A .. u I , .,u n

basis

g

inteser such that

For every

Let

S.

invertible

result

1.4.6 THEOREM.

be a group,

that for every

(IX G X A) U 0

~n+I,u(V)/~n,u(V)

(or full)

The following

sets, G

acting as zero.

n < dimV. For

n Xn

called the seneral

set

O.

Here

is called the sandwich matrix of

1.4.5 DEFINITION.

0

= (i,gp~jh,~)

instead of

called a Rees matrix semisroup.

semigroup

be nonempty with

PXi ~ 0, and for every

~°(I,G,A;P) (i,g,h)(j,h,D)

where we write

A

(~,i), be any function such

such that

P~i ~ 0.

and

G° = G U 0

_of _

of

V}, U} .

B, and for every

A E nU

fix a

Let

Gn,U(&,V ) = ~°(nu,Ln(~),nv;P ) where

P = (PBA)

with v1

=

.

PBA

A

is invertible, Qn,u(a,V)

let of

e

B A2 VlU

• ..

B nA I VlU

B A VnUl

B A VnU2

"'"

vBuA I n nJ

A

[ U l " ' " Un]

=

vB n

if this matrix

B AI vlu

and

PBA = 0

Finally,

let

= ~n+l,U(a,V)/$n,U(g,V),

be the function defined b_.x:

Gn,U(&,V )

otherwise.

e

maps the zero of

and

e:h ~ (b*u,[~ij],vb)

(h ~ ~,U(a,V) \ 0 )

Qn,u(g,V)

onto the zero

30

where

b*

is the conjugate

xb = Then

of

is an isomorphism i.

Let

and

~ b*U~ f. = ~ u. ~.. J i=l I ll

~ (xfj)v b j=l "''

PROOF.

b

of

~,U(~,V)

G = Gn,U(~,V),

for

i < j < n

onto

(i)

(x E V).

Gn,U(~,V ).

Q = ~,U(~,V),

and write

v1 BA [ViUj] =

.

A A [u I ... Un].

vB n In order

to prove that

have that

b*u E nU

have

b*U C_ U

Define

and

and

the

@

maps

Q

into

[~ij ]

is

invertible.

dim(b'u)

G, we must

= rank b = n

show that for

By 1.5

so that

and 2 . 5 ,

b C Q\0,

we

respectively,

we

b*U E n u.

functions

f. by t h e f i r s t formula in (1). ~4e w i l l show t h a t ] Vb Vb fl,...,fn form a basis of b*U. In fact, since v I ,...,v n is a basis of vb, vb vb f o r any x E V, xb i s a u n i q u e l i n e a r c o m b i n a t i o n o f t h e v e c t o r s v 1 ,...,v n Then

f.:V ~ A, and from the linesrity of b, it follows easily that each f. ] J linear. Thus f . E V* f o r 1 < j < n. F u r t h e r f o r any x E V, f E V , u s i n g l notation i n ( 1 ) , we o b t a i n

is

the

n x(b*f)

generated

as we have seen above,

follows

that

formula

in (i) implies

2.

= (~ (xf.)v\(b)f j=l J J

n ~ f (v~bf) j=l j by f l ' " ' ' ' f n '

b*f =

so that

V*

= (xb)f

that

let

have For

(the

b,c E Q \ O

and

by the

n

where T

in

Q

is

in

g(h,V))

= n.

Since

given if

T

of ~

is

where

vectors

form a basis Therefore

bf)),

dim(b'u)

b*U.

for

rank (bc)

It

Now t h e

Q

into

of

= n,

fl'''''fn"

maps

by:

the subspace

second

G.

b,c E Q'\0, = n

otherwise and

rank(bc)

[~ij ]

Vbc c Vc

and

dimVbc

b*c*U = b*U.

as in (i), and

n Vc xc = j=l >~, (xgj )vj , we then obtain

T

b*U ~ b ' V * c

fl'''''fn

product

=

b'V* c

{~ij] E Ln(A ).

the multiplication

Vbc = Vc; similarly f.j

that

is generated

and

that

0 Hence

T

b*U = b * v = T

Recall

follows

2 (xfj)(v j=l

Consequently

and

bc = { b c

we

It

=

n c*U gj = i~=lUi ~ij

for

i _< j _< n

(x 6 v),

= dimVc

= n,

31 n

c*U ]vVC (xb)c = J~--I[ (xb) ~ n ~ i~__lUi Yij ) j n n n b*u ]vVb~ n c*U Vc ~ [X(m~=~lum ~mk ) k j (if~=lUi Yij)vj j=l [ k=l ~

~ub,U

n n ~,. [ ~ . ( Z v , Vb u.c~U k=l mK i= 1 K i

x

j=l

j=l

m=l m

'

)~..]~ lj

vVC j

x( ~ u b*c*U , Vbc x(bc) m= I m °mj)Vj =

(2)

where we have set • Vb

c'U,

[oij] = [~ij ] IV i nj By hypothesis

rank(bc)

= n, so that

] [~ij ] .

(2) together

with

(i) yields

e:bc ~ (b*c*U,[oij],Vbc) since

the linear

forms f. and the matrix [~ij ] are unique. J ~ Vb c*U~ and thus also Iv i uj j and we obtain

is invertihle

(b~)(c@)

, Vb

c*U~

] [Yij ],Vc)

(b*c*U,[oij],Vbc) Suppose

next

that

1 ~ i < n, let

the hypothesis

rank(bc)

dependent. Hence, n XlbC = i~=2oi(xibc) with

rank(bc)

x.1

< n

Then

implies

bc = 0

in

V

with

for some

o i.

in

For

(bc)@ = 0.

xib = vVbi "

XlbC,...,XnbC

we may suppose

i _< j _< n, let

Then for

Q, so that

the property

that the vectors

vb c*U v i uj = (xib)(c*hj) and

= (bc)~.

loss of generality,

c*U c*h. = u . J J

the property

< n.

be any vectors

without

[~ij ]

= (b*U,[~ij],Vb)(c*U,[Yij],Vc) = (b'U, [~ij ] IV i uj

For

Consequently

h.j

Then

are linearly

that

be any vectors

in

U

i < i,j < n, we obtain -_ = (xibc)h j

hence

Vb c*U

v I uj which

implies

n n = (XlbC)h j = [i=2 ~ o.(x.bc)]h. i i J = i~_2°i(xibc)hj

of the remaining (b@)(ce) 3.

= 0

rows and thus

which

completes

To show that

(b*u,[~ij],Vb)

Vb c*U~ [v i uj j

that the first row of the matrix

~

r Vb

c'U,

iv i uj

j

that

is one-to-one,

let

= (c*U,[Yij],Vc)

with

is a linear combination

is not invertible.

the proof

[~ij] = [Yij], Vb = Vc, which

n ~ Vb c*U~ = i=2 ~ o.~v. i i u.J )

0

b,c E Q \ O

the notation

by (i) evidently

Consequently

is a homomorphism. and suppose

as above.

implies

that

Then b = c.

that

b*U = c'U,

32 4.

To show that

e

~n A f.j = =If~lu~iji

Then for Since

is onto,

for

let

[~ij ]

is invertible,

are linearly

[~ij] E Ln(~),

n B xb = J=~l(xfj)vj

i < j < n;

i <-- j _< n, we have that

fl'''''fn

A ~ nu,

f.j E A, which

it follows

independent.

then implies

the

first formula

there exist

in (3) and the calculation already

proved,

parison of (i) and (3), we deduce onto

that

of

b

is linear.

in (3) that

x. ~ V

in part i

the uniqueness

Define

(3)

that

from the first formula

Hence

B E n V.

(x E V).

x.f. = 6.. for 1 <__ j,i <__ n. But then by (3), x .b = v B. j i jl j j clearly vb ~ B, we obtain B = Vb and thus r a n k b = n.

From the statements

and

such that

Jso that

v B. E vb. Since j It further follows from

of the proof that [~ij ]

b8 = (A,[~ij],B).

b*U = A.

in (i), and a com-

Therefore

~

maps

Q

G. 5.

It remains

zero entry. a = aba

ab

to show that every row and every column of

a E Q \ 0, we may consider

for some

= rank(aba) both

For

b E ~u(V)

< rank(ab)

and

ba

by 3.6.

so that

rank(ab)

it follows

= n

of

Since

a

contains Su(V).

a non-

Hence

rank(ab) j r a n k a

and similarly

Q.

p

as an element of

By 1.3, we obtain

are nonzero elements

a(ba) = (ab)a = a E Q \ 0 ,

a

rank(ba)

= n.

is arbitrary

Thus

and

that the sandwich matrix has the required

property. 1.4.7 LEMMA.

If

S, then there exist PROOF.

a

and

c,d E S

b

are nonzero elements

such that

of a Rees matrix

Exercise.

We are now able to find all ideals and all principal Recall

semisroup

a = cbd.

that an ideal

I

of a semigroup

(or ring)

S

factors

of

is proper if

~u(V). I ~ 0, I # S.

The next result is new. 1.4.8 THEOREM. i) ~,v,(V)

If

dim V = m

I i < n < m}

and all principal ii)

Let

If

V

be a pair of dual vector spaces.

is finite,

then

ar___~erespectively

factors of

~u(V)

and all principal

in 4.6.

12 < n < m]

and proper ideals

{~n,u(V)

In = 2,3 .... }

the sets of all (distinct)

and proper

ideals

~u(V).

factor different

constructed

then

ar__~erespectively

factors of

Each principal

~n,v,(V)

the sets of all (distinct)

= gu(V) = g(V).

is infinite dimensional,

i Qn,u(V) In = 1,2 .... }

semigroup

(U,V)

from

QI,u(V)

is isomorphic

to a Rees matrix

33 PROOF. fications ~n,u(V) if

of the arguments is an ideal of

m ~ n Let

I

Qn,u(V)

be a nonzero a

and

b

to be presented

ideal of

F.

If

from 1.3, and that

a E F, b E I, and

elements

of

c,d E Qn,u(V) \ 0 ,

a = cbd E I, which implies ~n+I,u(V) \ ~ n , u ( V ) .

and is left as an exercise.

follows

Qn,u(V).

to the Rees matrix semigroup

for some

simple modiThat

~n,u(V)

# ~m,u(V)

of 2.4.

are nonzero

is isomorphic

a = cbd

the proof of part i) requires

F = ~D~u(V )

is a consequence

then both

have

We will prove part ii);

that

Next let

I

and thus

rank a = r s n k b

= n>0,

By 4.6 we know that

Gn,u(V ).

Hence by 4.7, we

c,d E F.

Consequently

is the union of some of the sets

b 6 1

and

rank b = n > m >

O.

Then

n b = __k~=l[ik'~k'~k ] with n minimal by 2.4, and v ,...,v k are linearly indeXl n pendent by 2.5. There exist u. E U such that _(vx ,u~ ) = Ok6kp for some 3p k Jp m 1 O k ~ 0 and i < k,p < n. Letting a = p~__l[jp,~p= ,Xp], we obtain n

m

.

-i

ba = (k~=l[ik,~k,~k])(p~=l[jp,Op

2 k=l p=l where both sets

[ik,~k(V

U.ll,...,Uim

2.5, we have that

rank(ha)

the above,

it follows

that

must have

Sn+l,u(V) ~

I.

unbounded,

then we have

Now let property:

J(a) = FaF). property

is

and = m

vhi,...,VXm and hence

Consequently,

O.

~n+I,u(V)

and hence

if and only if

of

The last statement

isomorphisms

with zero removed except

to a Rees matrix semigroup as a modified

for

constructed

replica

for some

It follows

Consequently F

Thus by

rankm.

of

I

n ~

2.

F

are

with

F

we

the

having

(since

this

that for any

l(a) = Sn,u(V)

containing

a

is given by

= Qn,u(V).

now follows [~u(V)

from 4.6.

as a tower of principal

factors

n = I.

Each of these factors

in 4.6,

so we can think of each of the

of the preceding

By

is arbitrary,

of higher rank

the only ideal of

factor of

of

m < n

is an ideal of

= ~n+I,u(V)/~n,u(V)

about

independent.

and no element

rank b = n.

I

an element

Since

J(a) = ~n+I,u(V).

In light of 2.6 and 4.8, we can view

"layers"

J(a)

rank n

that the principal

J(a)/l(a)

I.

I = ~n,u(V)

By the part already established,

in turn implies

Qn,u(V)

contains ~

E

if the ranks of elements

Then

an element

are linearly

I

~m+I,u(V) \ ~ m , u ( V )

I = F, otherwise

contains

b E F, J(a) = J(b) which

. ,u )o ,X ] = z.~ [ ' ,Xp] ~k Jp P P p=l :P'~P

a E F, rank a = n >

J(a)

,~p])

one.

is isomorphic

34 1.4.6 EXERCISES. i)

Find all ideals of

ii)

Characterize

iii)

g~(V)

and of

all one-sided

For a dual pair

£ (V).

ideals of a Rees matrix semigroup.

(U,V), show that the sets of all minimal right ideals of

the following semigroups and rings coincide:

b) ~.~u(V), iv)

c) ~u(V),

d) gu(V),

For any subspace

a ~ £(V),

show that

V'

a) Sn,u(V)

of

V, let

~(V)a = Xva(V ). ~:V ~ ~ Ev~(V )

left ideals of

is isomorphic

£(V).

~v,(V) = ~b E ~(V) IVb ~ V'}.

(V l

is a subspace of

to the lattice of all principal

Throughout

this section,

l,

For

V

V) onto the lattice

of all

Deduce that the lattice of all left ideals of

1.5

spaces.

n >

Prove that the m a p p i n g

is an isomorphism of the lattice of all subspaces of principal

for any

e) ~u(V).

left ideals of

$(V)

~(V).

SEMILINEAR ISOMORPHISMS

let

(U,g,V)

and

(U',h',V ~)

be pairs of dual vector

Our purpose now is to express every isomorphism of a ring

~

onto a ring

~', where

~u(~,v) ~

~u,(&',v') ~ ' ~£u,(a',v'),

C£u(~,v).

in terms of mappings of respective pairs of dual vector spaces. 1.5.1 DEFINITION. ordered pair

(w,a)

A semilinear

where

additive h o m o m o r p h i s m

w

(written on the right) of

(ov)a = (ow)(va) If

a

is one-to-one and maps

are said to be isomorphic, In the case that

instead of

(w,a)

V

(~,a)

V

(h,V)

into

onto

into

At

V'

(AI,V j)

and

a

onto

V',

and

(~,a)

such that

is called a semilinear isomorphism

w

(&,V) ~

(g,V)

and

(&',V')

(&',Vt).

is the identity automorphism,

is a linear transformation.

is an

is an

(i)

g = 4', V = V'), and then

to he denoted by

& = &~

h

(o E 4, v E V).

(semilinear a u t o m o r p h i s m if also

transformation

transformation of

is an isomorphism of

the semilinear

We will usually write

w h e n there is no danger of confusion.

a

This is also justified by

the following statement. 1.5.2 LEMMA.

Let

a

be a nonzero additive h o m o m o r p h i s m o f

there exists at most one function exists,

it is a ring homomorphism.

w:h ~ ~' If

a

satisfying

V

into

(I), and if such an

is one-to-one,

then

~

is also.

V~

Then

35

PROOF. then

for

= w'.

Let

x E V

that

any

o E 4, we h a v e o,~ E 4, =

(o7)w =

and h e n c e

o~ = ~w,

that

then

It f o l l o w s function trivial

of

can also

right

vector

Let

[(o+~)x]a

=

(o~)(xa)

that

spaces

(U,&) b(u~)

=

or

b

and

into

Let

a function

semilinear

this

is

LEMMA.

If

b = c

transformation

Let

(v,bu ~) = and

o~ = ow'

and

(i),

thus

(o~)(~w)(xa)

(dx)a +

(Tx)a

homomorphism.

=

(~x)a

of

is

a

so

If

a

ox = ~x

is o n e - t o - o n e which

w

implies

we w a n t

U,

the

be a o n e - t o - o n e

transformation

is a

the o n e - t o - o n e n e s s

to show

of a s e m i l i n e a r

is dual,

(u E on

that

the zero

is o n e - t o - o n e , when

(Ut,& I)

that

a given

transformation

in p a r t i c u l a r

of

of

(i) b e c o m e s

~ E 4)

(2)

left).

be o u r

dual

pairs

and w r i t e

instead

(v,u)

~(v~,u').

(m,a)

be a s e m i l i n e s r

transformation

is an a d j o i n t

of

(w,a)

= (va,u~) '

(v ~ V, u ' ~ U ' ) .

a generalization

of the definition

(~,a)

c

of

if

b

has

=

(v,cu ~)

the a d j o i n t

an a d j o i n t

(U~,& ~) is o n t o

be a n o t h e r

(v,bul)w so that

satisfy

of

(g,V)

into

if

(3) o f an a d j o i n t

given in

i.

is on___~e-to-on___~e, and PROOF.

a

definition

b:U ~ ~ U

(v,bu')~

b

w ~

(o~+~w)(xa)

to r e q u i r e

( U I , & ~ , V ~ ; ~ ~)

DEFINITION.

1.5.4

(Tw)(xa)

that

(written

1.5.3

(2) of S e c t i o n

=

(bu)(~w)

instead

Observe that

and

that

=

=

=

is a r i n g

is u s e f u l

The

(vl,u~) '

Then

(ox+Tx)a

(~)(xa)

case w h e r e

lemma

~(v,u),

(&~,V~).

=

it s u f f i c e s

in the c a s e

This

(U,&,V;g)

(ow)[(~x)a]

w

(ow)(xa)

(the

and

(b,w)

so

=

+

Thus

is s e m i l i n e a r .

and we w r i t e

of

[o(~x)la

=

=

~

be o m i t t e d .

transformation

(owJ)(xa)

~

is o n e - t o - o n e .

onto

exception),

=

If b o t h

and

(ox)a ~

from 5.2

B

=

= o~ + ~w.

~ = T; h e n c e

xa # O.

(ow)(xa)

[(o~)x]a

(Ow)(7~),

(o+T)w

that

then

[(o+~)~](xa)

and

such

Let

[(o~)®](xa) so

be

into then

adjoint

(va,ul) I = for all

is u n i q u e .

b,

of

it is u n i q u e

(U,&). a

if

(b,w a

-1)

is a

is o n t o

then

is o n e - t o - o n e . (~,a).

(v,cnl)w

Then

for any

u ~ E U ~, we have

(v E V)

v E V, w h i c h Further,

Moreover,

and

implies

that

bu ~ = cu ~.

Hence

36

(v,b(u'+s'))~

= (va,(u'+s'))' =

t (v,bu)w

+

= (va,u')'

(v,bs')~

+ (va,s')'

s [(v,hu ~ ) + ( v , b s ) ] ~

=

= (v,bu'+bs')w which implies b(u ~ +s')

that

(v,b(u ~ +s'))

= b u ' + b s ~.

= ( v , h u ~ + b s j)

= (va,u%)'

= (va,u')'o

= [(v,bu')(om-1)]m implies

b(u'o)

=

that

If

a

maps

A similar

This

=

onto

V'

(h,w -I) and

and t h u s

= bs', (va,se) j

(va,u')'

proof

shows

that

if

lemma

shows

that

by d u a l i z i n g

b

(b,m-l),

If

a

for

all

v E V

is a s e m i l i n e a r

=

that

of

= [(v,bu')~][ow-l~]

bu'

so

an a d j o i n t

v E V

= (v,(bu')(om-1))~

(v,(bu')(om-l))

Therefore

V

(v,bs')

is also

(v,b(u'~))

(bu')(ow-l).

(v,bu ~) =

all

Also

(v,b(u'o))~

which

for

is onto,

then

for any

which

then

a

the c o n c e p t

or s i m p l y

that

a

implies

that

u ~ = s ~.

be o n e - t o - o n e .

of a d j o i n t , and

thus

v E V, we h a v e

then

must

and

transformation.

h

are

we m a y

say

adjoints

that

a

of e a c h

other. 1.5.5 one-to-one

LEMMA. function

of

is an a d d i t i v e ~

onto

h',

b

homomorphism

of

is a f u n c t i o n

V

onto

mapping

U'

V', ~

is a

onto

U,

such

that

then

lemma

(~,a)

is

(o E 4,

(v,bu')~ = (va,u')'

(v E V, u ' E U ' ) ,

a semilinear

PROOF.

This

In this

lemma

is s s p e c i a l

Clifford

(ov)a = (O~)(va)

follows

isomorphism of

from 5.2

the h y p o t h e s e s case

and P r e s t o n

of

(&,V)

on

a

and

the h o m o m o r p h i s m

([i], C h a p t e r

(5)

(&',V')

b

may

theorem

c:I I ~ G, w i t h

3, S e c t i o n

that

and o n t o

matrix

adjoint

b.

isomorphism

d:A ~ G ~, w i t h

d:X - dh, and

and

onto

The

following

semigroups,

see

4). the R e e s

matrix

(written

on

the

left),

the

right)

O~ = 0 ~,

(written

semisroup

S' = ~ ° ( I I , G J , A I ; P I ) .

c:i I -- ci, ,

w : G -- G ~, o n t o

~ : A ~ A I, o n e - t o - o n e

with

be i n t e r c h a n g e d . for R e e s

functions: ~ : I I ~ I, o n e - t o - o n e

such

onto

(4)

and 5.4.

1.5.6 LEMMA. Let 0 be an i s o m o r p h i s m of o S = ~ (I,G,A;P) o n t o the R e e s m a t r i x s e m i s r o u p exist

v C V),

on

Then

there

37

(i,g,X)~ = ( -li,(c ~lig)~d%,X~)-I

((i,g,~) E S),

(6)

d ~ p ~r, i ~ = (p~,~iJci~)~

(X E A, i

(7)

PROOF. form

First note that the nonzero

(i,p$~,~)A~ where

to be denoted

Phi ~ O.

i, such that

(l,pi~,l)6 = (k,PSkl,~)

Pll ~ 0.

for some

idempotents

of

Fix an element in Then

S

A

are the elements of the

and an element in

(l,Pl~,l)

k E I l, ~ E A~.

E I~).

is an idempotent

Any element

(i,g,X)

I, both and hence of

S

can be written in the form -l -i (i,g,~) = (i,e,l)(l,Pllg, l)(l,Pll,X) where

e

is the identity of

G.

There exists

D 6 A

(8) such that

P~i # 0, so

(k,P~kl,') = (1,P;~,l)~ = [(1,Pi~g-lp~,~)(i,g,1)]~ = (l,p;~g-lp~,.)~ (i,g, 1)~ which shows that form

(k, , ).

(i,g,l)~

is of the form

( , ,~); similarly

(l,g,X)~

Using this and motivated by (8), we define the functions

q:i ~ qi,o), d:% -- dx, ~

is of the ~,

by

(i,e,l)~ =

(~i,qi,'r)

(i E I),

(l,Pllg,l)~ = (k,P~k (gw),~)

(g E G°),

-1 (l,Pll,X)~

(~ E A).

= (k,P~kldx,~)

Then by (8), we obtain (i,g,X)e = (i,e,l)~(l,pi~g,l)@(l,pi~,X)~ = (~i,qi,~)(k,PSkl(gw),~)(k,PSkldx,X~)

(9)

= (~i,qi(gw)dx,X~). Further, by the definition of -

(k,P~kl(gh)m,~)

~, we have

~:G ~ G ~

and

0w = Or; also

-i -i -i = (l,Pllgh,l)0 = [(l,Pllg,l)(l,Pllh,l)]0

= (l,Pl~g, l)e (i, Pl~h, I) e = (k,PSkl(g~),~)(k,P~kl(h~),~) = (k,P~kl(g~)(h~),~) which implies

(gh)w = (gw)(hw).

definition of

~, we have

If

g~ = e ~, the identity of

-i ~-i ~-i -i (l,Pllg,l)0 = (k,P~k (g~),~) = (k,P~k ,~) = (l,Pll,l)~ so that

g = e, and

•

is one-to-one.

G j, then by the

38 Using

(9), we see that for any

(k,qladl,~) maps

G

= (l,g,l)~.

onto

For

Hence

(j,e,~) E S ~, let

implies If

that both

= (k,ql(gW)dl,~)

such that

so

a = g~

and

(i,h,~)

= (j,e,~)~ -I.

= (i,h,X)@

~

and

~

We obtain by (9),

= (~i,qi(h~)d%,X~)

are onto.

~i = ~j, then again by (9), we have (i,e,l)~

which

(k,qladl,~)

g E G

G j.

(j,e,B) which

a E G ~, there exists

implies Further,

= (~i,qi,~)

that

i = j.

using

(9) we obtain

(i,e,~)20

-i -I -i = (j,(qj qidl )w ,l)e

= (~j,qj(q]lqi),T)

Consequently

= (i,P~i,~)0

~

is one-to-one,

= (~i,qi(P~iw)d

property, =

p%i w Finally, functions

~

letting and

~ =

c

d

~

is also.

,~) ,

[(i,e,X)~] 2 = (~i,qidx,XTl) 2 = (~i,qidxP i so by the homomorphism

similarly

~,~ iqid~,X~)

we have

xpx~, ~iqi •

g-l

and

(i0)

-1

ci~ = q ~ i ~

as in the statement

-i

for all

of the lemms,

i ~ E I ~, we obtain

the

and (6) and (7) follow

from (9) and (i0), respectively. 1.5.7 DEFINITION. a semilinear b

which maps

U~

to be isomorphic, Strictly

A semilinear

isomorphism onto

U.

The dual pairs

and we write

speaking we should write

not necessary

by 5.4.

We will

fusion.

Again by 5.4, we know

(U~,& ~)

onto

Then

(~,a)

(~-l,a-l)

the function

~(~,a) c~(~,a)

is an isomorphism PROOF.

o__f £U(&,V)

Exercise.

a

instead

-- (&~,V ~) (~,a)

(U I ,~ I ,V ! )

snd

is called

has an adjoint are then said

(UI,&~,V~).

(b,m,a)

of

that then

and hence

Let

(U,£,V)

if

instead

of

since all we need is the existence

also write

(U,£),

1.5.8 LEMMA.

(U,&,V),

onto

(U,&,V) ~

(w,a):(A,V)

l l (U I ,g ,V )

(U,&,V)

actually

(UI,~I,VI).

isomorphism

of

(m,a) (b,m -I)

isomorphism

be a semilinear

b

but this is

which

is then unique

if there is no danger of conis a semilinear

of dual pairs

is ~ semilinear

(~,a),

of

isomorphism i>omorphism

isomorphism

is an equivalence of of

(U,&,V) (UI,&I,V ~)

onto onto

defined by = a

-i

ca

(c E £U(~,V))

ont_____o £U~(&

,V ), and

of

relation.

rank c = rank(c~(~,a) ).

39 We wish

to show that every isomorphism

of a semigroup

S

onto a semigroup

S ~,

for which $2,U(h,V) with

dim V >

isomorphism

several

important

(~,a)

of

dim V = I.

Let

of Section

O

8

/ P%~i'

/ = (v%,

define

,Iv;P ) i )'

functions

Then

(w,a)

a

and

b

i.

then

Since

of the general

S = $2,U(&,V)

case;

onto

E V, u i E U ~, v ki fi V I

and

S~

as Rees matrix

be as i n 5 . 6 .

as

at

~h__!e b__e_e-

semigroups

O

t-

Suppose

that

S ~ = ~ (Iu~,~&

and

.

,Iv~,p

J

)

with

dim V >

1, a n d

b_~

= u ici(Tw - i)

i C U'). (ui~

isomorphism

of

E V),

(U,~,V)

(ii) (12) onto

(U',~/,V ~)

with adjoint

= ~, of

the

first

the representation

Ow = 0 1

~

for

proves

2.

S

of

b(u~T)

implies

= (~w)d v ~

statement

consists

of

of nonzero

elements

a verification

of

the

= ~

b

0a = 0 I, a

# 0 I.

so that

It follows

~ = 6.

of

V

in the form

is single valued. that

We thus have

~

If

= B~

~v

=

and hence

~v

and

?v X

that

a

k~ _l]a maps

is defined

one sees that For any

a

k = ~.

But

is one-to-one.

= [(~d ~ -I- l)w-lw]d V

onto

i v~k ~ -~~ =

~v~~

(13)

V'.

in a way completely b

~-

is a one-to-one

analogous function

to that of of

U~

onto

a, so by a s~milar U.

o E ~, ~v k E V, we obtain by (ii),

[o(~vk)]a = [(~)vk]a = (o~)~dkv'k~ = (°~)[(v~)d~vk~ ] = (o~)[(~v)a] which verifies For any

is

(~vx)a = ( 6 v ) a # 0,

0 ~ ~ Wv~ E V ~, we obtain

/(~d k~--I l)w-lv

Now

false

V)

(vw)dxv~

argument,

for some semi-

then establish

in 5.5.

then also

which

for the treatment

(~v

The p r o o f

and

Further

{(w,a)

We will

and show that it is in general

= (~)dkv~

I$2, U (~,

unique,

(U',A',V').

(~v)a

b and ¢(w,a)

hypotheses

u i E U, v both

~,c,~,d,~

is a semilinear

PROOF,

is crucial

Phi = (v~ ,ui)

with

and l e t

U I

of an isomorphism

be an isomorphism

2, consider

-

S = ~ (Iu,~&

~ S' ~ gU,(h',V')

[4].

= $2,U ~(~ ' ,V ' ), fix vectors

$innin$

S

onto

of this result

The next result

1.5.9 THEOREM.

to

(U,h,V)

consequences

it is due to Gluskin

S'

$2,U,(£',V')

i, is the restriction

linear

for

c S c gU(&,V),

(4). ~vx E V

and

u~l E U ~, we obtain

by (7),

(ii),

and (12),

40

(7vk,b(u~T))w = (~vk,u ici(T~-l))w d

= (~) which verifies

3.

Let

(xia,b

-i

Xl,X2,X 3

a

is additive

be linearly

such that

and will do this in several

independent

(xi,gj) = 6ij

gj)1 = (xi,gj) ~ = 6ijw = ~ij

for for

vectors

in

1 ! i,j < 3. I < i,j < 3. -

for

a

/

,uiT)'

(5).

We will now show that

gl,g2,g3 E U

= (3~0)[(vk,u i)ci]wT

~ J ~T I u I x(vX~,ui ) = ((~w)dxvx~ , iT) ` = ((~vx)

V.

steps.

There exist

By (5), we have 3 If ~ o.(x.a) = 0, then

--

i=

1

z

i

i < k < 3, we have 3

3

-1 o k

and hence 4.

= Ok(Xka,b

xla,x2a,x3a Now let

x

-

gk) =

~ o.(xia,b i=l I

are linearly

and

y

-

lgk) = (i~__lOi(xia),b

igk) = 0

independent.

be linearly

independent

vectors

in

V.

The vectors

xa,ya,xa+ya are linearly dependent so by part 3 we have that the vectors -i -i -i xsa ,ysa ,(xa + y a ) a are linearly dependent. Since x and y are linearly -i independent, we must have (xa + y a ) a = ox+Ty for some o,T E 4, whence xa+ya Using

= (ox+Ty)a.

(14) and (5), we obtain for any

(14) u I E U ~,

(x,bul)w + (y,bul)~ = (xa,ul) I + (ya,u')' = (xa + y a , u l ) ' = ((Ox+~y)a,ul) I = (Ox+~y,bul)~ There exists yields

u E U

lw = o~.

such that

=

[O(x,bu I) + T ( y , b u l ) ] ~ .

(x,u) = i, (y,u) = 0; letting

Consequently

o = i

and analogously

(15)

u ~ = b-lu,

T = i.

Formula

(15)

(14) then

becomes (x+y)a for the case in xhich 5.

Again

let

x

z

are linearly

It follows easily

so that

xa + y a

vectors

x,y.

y

(x,y E V) are linearly

If

x = 0

but both nonzero,

independent

that if

linearly independent. xa+ya+

and

x,y E V.

are linearly dependent and

= xa+ya

x+y

Using

= (x+y)a,

y = 0, (16) is trivial. z

be a vector

(here we are using

(16) several

which a

let

independent.

~ 0, then both sets

za = x a + ( y + z ) a

Therefore

or

(16)

implies

V

the hypothesis x,y+ z

and

If

x

such that that x+y,z

and

y

y

dim V >

i).

are

times, we obtain

= [x+ (y+z)]a

is additive.

in

= [(x+y)+

that (16) is valid

sin = ( x + y ) a + z a for linearly dependent

41 By 5.5, (m,a) adjoint 6.

is a semilinear isomorphism of

(U,~,V)

onto

It remains to verify the last statement of the theorem.

[i,~,~] ~ 32,U(~,V )

and

with

= ~(od-i l)~-iv = od-l-i [(v

= [(ov~)a-l} [i,~,X]a

_i } [i,v,k]a = [(od -I_I)~-I( v

-l'Ui)~}wdkvk~ = od-l-i (p

_l,Ui)~vx}a

-i .~)(~)dkvk~

-

= od -I i d

~-

~

_i p'

-i

_li(C !li~)wdkv~

~,~

~(~,a)I~2,U(~,V)

i

= o(v',u' , )'(c _li~)~dkvkl

~

= (ov~)[~-li,(c~!liV)~dk,k~] which shows that

~-~i

= (~v~)@

= @

and completes the proof.

1.5.10 DEFINITION. Let S be a subsemigroup of gU(g,V) U ~ { i ,V i). An isomorphism ~ of S onto S i

and

semigroup of

by a semilinear isomorphism =

4'

For

ov I~ E V', using (13), (ii), (7), (6), we obtain

(ov~)(a-l[i,?,X]a)

~(~,a) Sl

(UI,~',V I)

b.

(~,a)

of

(U,g,V)

onto

S'

be s sub-

is said to be induced

(U',&',V ~)

if

This same definition is to be used for subrings of

£U(~,V)

and

£U,(8',V'). In particular 5.9 says that every isomorphism of is induced by a semilinear isomorphism.

$2,U(h,V)

onto

$2,U,(& ,V ~)

We will show that the conclusion is valid

under much less restrictive circumstances. 1.5.11LEMMA. 32,u(V )

If

is a subsemisroup of

gu(V)

containing

~2,u(V),

is the unique nonzero ideal contained in every nonzero ideal of

PROOF.

Write

52 = $2,u(V).

52 ~ S ~ ig(v)(32) S, and let

and thus

a C I\0.

ba E 32 N I. But then

S

For any

c E I

52

By 2.8, part i), the hypothesis is an ideal of

S.

By 3.3, there exists

b ~ 32

c C 32 , there exist

g,h C 32

and hence

Let such

I

S.

implies that

be a nonzero ideal of

ha # O; also

such that

c = g(ba)h

For semigroups of matrices,

1.5.12 THEOREM. Sl

Let

be a subsemigroup of

dim V > i, and let

~

by 4.7.

32 ~ I.

We are now able to prove the main result of this section, due to Gluskin

guise, and then by Gluskin

then

this was first proved by Halezov

[4].

[2] in a different

[3]. S

be a subsemigroup of

gUI(&I,V ')

containing

be an isomorphism of

S

gU(&,V)

containing

32,U,(~',V'), onto

S ~.

Then

$2,U(g,V),

suppose that ~

is induced by a

42 semilinear isomorphism

(o0,a) of

unique extension of

to s homomorphism of

PROOF. ideal of

~

~2,U~(f~,V~),

Letting

hypotheses of 5.9, so

~

i ~ IU

$2,u(A,V) S.

gu(A,V)

V~

into

is the unique nonzero

must map e

gU~(A~,V ~) (~)dxv~

V ~, and this representation is unique since such that

~

we see that

in the form

(vX,u i) = o ~ 0

so that

is the

Since the same kind of

a

~2,u(A,V)

satisfies the

is induced by a semilinear isomorphism

q0 be a homomorphism of

onto

and ~(u~,a) guI(A i ,V J).

into

it is clear that

e = ~I~2,u(A,V )

write an arbitrary element of V

(U',AI,V l)

gu(A,V)

contained in every nonzero ideal of

~2,uI(A~,VI).

Let

onto

First note that by 5.11, we have that

S

statement is valid for onto

(U,A,V)

(w,a).

which extends

~.

as we may since is one-to-one.

~v X = (~vx)[i,o-l,kl.

a

We maps

There exists

Hence for any

c ~ gU(A,V), we obtain [ ( ~ ) d k v ~ ]a-lca = (~vk>ca = [ (~v k) [i,(~-l,k]}ca

(~vx)a[a-~([i,o-~,X]c)a} = (Vv~)a{ [i,o-~,X]c}~

= t (Vv~) [i,c-~,~l}a(c~) = (~v~)a(c~) = { (w)d~v~J (c~) and thus

cop = a

-I

ca.

Consequently

same calculation with Since

cp

is the unique extension of

~

to a homomorphism of

it is also the unique extension of

Defining

£(&1,Vl).

contained in

~(w,a)

~(~,a)

~U(&,V), we see that

~(~,a)

~UI(&I,V I)

if

subsemigroup of

U~

~

gu(A,V)

containing

into

to such a homomorphism.

~

~U(&,V)

onto

to an isomorphism of

is contained in

~(~,a)

to s, the ~(0~,a)IS = ~'

by the same formula as on

SU(~,V),

preserves rank, and Ii~u(&,V )

are t-subspaces of

~2,U,(&,V)

then

Sl

~(&,V) is

~(~,a)I~U(~,V) into

is

[~UI(&~,VI).

V*, then every isomorphism of a

~2,UI(&,V )

onto a subsemigroup of

can be extended to an (additive and multiplicative)

~(~,V).

1.5.13 COROLLARY. be a sabring of

£(&,V)

to a homomorphism of U~

8UI(g,V)

containing

automorphism of

since ~

and

~

is an extension of S

e

c ~ S, shows that

is in fact an isomorphism of

on all of

For example, if

the unique extension of Furthermore,

let

extends

and for

~e know by 5.8 that

gU,(~,V)

~

~0

£UI(~J,VJ).

onto

q0 = ~(0~,a)" Since

instead of

gu~(AI,V~),

~

Let i

i

~Ut(~ ,V )

be an isomorphism of

~

be a subrin$ of i

containing ~

onto

5.12 and is thus a ring isomorphism of

£U(g,V) i

containing

~UI(~ ,V ), suppose that rD.~~. ~

Then

onto

~.

~

~U(g,V),

~

dim V > i, and

satisfies the conclusions of

43 The ring version restriction

of 5.12, with

on the dimension of

the rings

~l = ~u,(V) ' the result was established

isomorphism

every multiplicative cannot be dropped. ~(V') ~ 4'.

~'

as in 5.13 and no

[3]; for

the remarkable

~ = Su(V),

[2].

property

that every

of a ring in that class is automatically additive,

isomorphism For if

is a ring isomorphism.

The hypothesis

dim V = i, it is easy to see that

But there exist division

multiplicative

and

earlier by Dieudonn~

In 5.13 we have a class of rings with multiplieative

~

V, is due to Jacobson

isomorphisms

which

rings

(even fields)

are not additive.

i.e.,

dim V >

i

~(V) ~ 4; similarly

for which

For example,

there exist

let

4 = 4~

be

the field of real numbers and n be any odd integer, n # i; then the mapping n is a multiplicative automorphism of ~ 4 but is not additive. It follows

x ~ x

that the corresponding isomorphism which

automorphism

in turn implies

of

£(V)

that

is not induced by a semilinear

dim V >

1

cannot be dropped in 5.12 or

5.13. For

0 < ~ _< (dim V) +, let

1.5.14 COROLLARY. that

PROOF.

Then

If

~

Conversely and

~

i) ii) iii) iv)

v)

4

and

if

of

4'

g ,U(g,V)

is induced by a semilinear

(w,a)

¢(w,a)

preserves

is a semilinear

~ = 4 ' , then

be cardinal numbers

statements

onto

such

__if --and onl~ __if S = 4 ~

g4,,U,(4~,V~),

isomorphism

rank

(~,a)

we must have

isomorphism of

and

(4~,V ~)

then by

of

(U,4,V)

~ =

(U,4,V)

is the required

~(~,s),l~

4, U(g, V) For vector spaces (A,V)

1.5.15 COROLLARY. the following

and

g4,U (4'V) ~ ~s',U ;(g~'V~)

is an isomorphism

(U ,& ' ,V ' ), and since

(U ,4 ' ,V ' )

i

fl ~U(4,V).

(UJ,4J,V').

5.12 we know that

onto

isomorphism. with

dim V >

i,

are equivalent.

& ~ 4', dim V = dim V' (4,V) ~

(4~,V').

g (4,V) ~ g

4

~(4',V ~)

for some cardinal

numbers

Q,4 ~ >

i.

g(~,V) ~ g(a',V').

£(4,v) ~ £(a',v').

PROOF. one-to-one a

dim V >

i < 4,s' < (dim V) +.

an__~d (U,a,V) ~

onto

Let

g4,U (&'V) = ~4(&'V)

i) = ii).

Let

function mapping

to all of

V

w

be an isomorphism

a basis

B

of

V

of

4

onto

onto a bssis

by defining n

va = k~=l(~k~)(VkS)

n if

V = k~=l~kVk

(v k E B).

~ B~

and of

V ~.

s

be a Extend

44 Verification

that

a

is a semilinear isomorphism of

(~,V)

onto

( ~ , V ~)

is left

as an exercise. ii) = iii).

This follows

iii) = iv).

This follows from 5.12 with

iv) = v) and v) = i).

from 5.14. U = V*, U ~ = V ~*.

This follows from 5.13 with

This corollary says that under the hypothesis determines

(~,V)

the semigroups

and the ring

We now investigate when two semilinear (or ring) isomorphism. of

G

induced by

If

G

g, i.e.,

1.5.16 DEFINITION. (x E V)

(~,V).

V

Let

m

(&,V)

U

of

V*

then

a

PROOF.

Let

a

defined by

xm

= ?x

is a semilinear auto-

but most of the proof is old.

be a vector space.

If

is s multiplication.

i E IU

the hypothesis

dim V,

?, or simply a multiplication.

a

is any function $2,U(&,V)

Conversely,

for some

every multi-

£(V).

be as in the statement of the theorem.

and there exists

~2,u(V),

and

the inner automorphism

into itself which commutes with all elements of

t-subspace

g

g

~ ~ O, m~ = (~ _l,m )

~ l i c a t i o n commutes with all elements of

x = ~vl_

i, any of the following

induce the same semigroup

~ ~ &, the transformation

The next result is new,

1.5.17 THEOREM. mapping

in

isomorphisms

c

It is easy to verify that for

U ~ = V ~* .

the division ring

is a group, we denote by -I xc = g xg (x E G). g

For

and

~(~,V).

is called the m u l t i p l i c a t i o n induced by

m o r p h i s m of

dim V >

up to a semilinear isomorphism:

g (~,V), g(~,V),

U = V*

such that

Let

(vX,ui) = o # O.

0 ~ x E V. Since

Then

[i,o-l,k]

is

implies

[(xa,ui)o-l~-l]x = (xa,ui)o-lvk = x(a[i,o-l,~]) = x([i,o-l,k]a) = (~(vk,ui)o-lvk)a = (~vx)a = xa. It follows that

a

some scalar

we have

~x

there exists

maps

b E ~2,u(V)

x

into the subspace of

xa = ?x x.

If

such that

x

and

xb = y.

V y

generated by

x, and thus for

are nonzero vectors,

by 2.10

Using the hypothesis, we obtain

~yy = ya = (xb)a = x(ba) = x(ab) = (xa)b = (~xX)b = ~x(Xb) = VxY and hence x # 0. quently

~x

Let

=

o

a = m

Conversely,

~y.

But then

=

~x

be the zero function on

is a constant and we have V.

Then

if

b ~ £(V),

m b = bm

.

then for every

x E V, we have

= (~x)b = ~(xb) = ( x b ) m

xa

=

~x

0 = (0a)o = (Oo)a = 0a.

•

x(m b) = ( x m ) b and thus

~

= x(bm~)

for all Conse-

45 If C(S)

S

is a semigroup

[a ~ S I xa = ax 1.5.18 LEMMA. PROOF.

(or ring), by

for all

For

C(S)

we denote

the center of

S, i.e.,

x E S} .

~ ~ &, m

is linear if and only if

~ E C(&).

Exercise.

1.5.19 COROLLARY. PROOF.

If

a E £U(g,V),

C(£U(g,V))

=[m

IV E C(~)} .

a ~ C(£U(~,V)) , then

by 5.18 we also have

by 5.18, we have

m~ E £(g,V),

£(~,V).

for any

Further,

a = m

that

for some

~ E C(~).

and by 5.17, m

? E ~

by 5.17,

Conversely,

let

and since

~ E C(A).

commutes with all elements

Then

of

x E V, f E V*, we obtain

x(m~f) = (xmv)f = (vx)f = v(xf) = (xf)v = x(fv) which proves

that

m*f = f~

again a multiplication). and therefore

m

( ~ , V ~)

If

and of

a semilinesr

Consequently

(~,a)

onto

~ E & . PROOF.

( ~ , a ~)

into

of

Let

into

Then

that

m v E £U(&,V)

transformations

respectivel~,

then

of

(~0o0 ,as )

(&,V) is

and

( ~ , a ~)

The equation

a~a -I

-i

~(w,a) = ~(w~,a ~)

ca = a'-ica '

to

isomorphisms

if and only if

of

aI = m a

for

to

to

as

a~a -I = m

a ~ = m a.

is equivalent

(c E ~U(&,V))

commutes with all elements

is equivalent

be semilinear

I_~n suc____~h~ case, w ~ = ¢ _i w.

in turn can be w r i t t e n

equivalent

implies

is

(g~,VH).

~(~,a) = ~(~i,al)

(a'a-l)c = c(a' a -1) i.e.,

which

are semilinear

(&~,V~),

(A,V)

(~,a)

(UI,&I,VI).

a which

and

( ~ , V ~)

transformation

1.5.21 COROLLARY.

some

m*U c-- U

of a multiplication

Exercise.

PROOF.

(U,&,V)

the conjugate

E C(£U(&,V)).

1.5.20 LEMMA. into

(in particular,

for some

(c E £U(&,V)), of

~U(&,V).

~ E A-;

The last statement

(17)

Hence by 5.17,

the last expression

of the corollary

formula

(17)

is evidently

follows

from 5.20.

1.5.22 EXERCISES. A ring

(~,+,.)

is said to have unique addition

if for any ring

defined on the same set and under the same multiplication, This concept was introduced i)

by R.E. Johnson

Show that a ring

every isomorphism

of

~t

that

(~,-) + = ~.

[i].

has unique addition onto

we have

is additive.

if and only if for any ring

~,

46 ii)

A ring

~

all of whose elements are idempotent is called a Boolean rin$.

Show that a Boolean ring has characteristic

2, is commutative~

and has unique

addition. iii)

Show that the ring

Z

of integers has only one automorphism,

has an infinite number of automorphisms.

Hence

Z

wheras

~Z

is not a ring with unique ad-

dition. For any ring iv)

Let

~

~, by

3+

denote the additive group of

be a ring and

F

3.

be a prime field (i.e., F

is either the field

of rational numbers or the ring of residue classes of integers Prove that if ~ 2 = 0.

not additive F+

[F0~~ ~F,

then

~ ~ F, and if

Give an example of a prime field (and hence

F

mod a

~ + ~ F +, then either F

prime).

~ ~ F

w i t h an a u t o m o r p h i s m of

does not have unique addition)

or

~F

which is

and an automorphism of

which is not multiplicative. v)

Prove that if

~

is a ring for which

3 + ~ Q+/Z +

rational numbers over the additive group of integers), example of a semigroup vi) g(V)

S

Show that for a vector space

and any nonzero ideal of An alsebra

over a field (A,+,.)

4

then

with zero and with the property

(g,*,A,+,.)

V

~(V)

with

dim V >

(the additive group of ~ 2 = 0. S ~ ~0~

with scalar m u l t i p l i c a t i o n +

i, any nonzero ideal of

*

and addition

and m u l t i p l i c a t i o n

~(ab) = (~a)b = a(~b)

etc.

+, and a ring

(~ E 4, a,b E A)

dim V >

Let i.

(U,4,V) Show that

A.

be a pair of dual vector spaces where ~U(4,V)

x(~a) = ~(xa)

algebra i s o m o r p h i s m of inner a u t o m o r p h i s m of

(U,4,V) ~U(4,V) £(4,V),

4

can be made into an algebra over

is a field and 4

by defining

(x E V, ~ ~ 4, a E £U(4)).

prove that every algebra isomorphism of linear isomorphism of

The concepts of

for an algebra refer to tbe same notion for both the

vector space and the ring structure of vii)

(4,*,A,+)

., and satisfying

where we have denoted both "multiplications ~' by juxtaposition. homomorphism,

~.

have isomorphic a u t o m o r p h i s m groups.

is a system consisting of a vector space

with the same addition

subalgebra,

Give an for no ring

onto onto

~U(g,V)

(UI,4,V~). £U~(4,V )

onto

~UI(4,V~ )

Deduce that for

is induced by a V = V l, every

can be extended to an algebra

and that every algebra a u t o m o r p h i s m of

£(&,V)

is

inner. viii)

Prove that if

4

is a field,

leaves the elements of the center of

then every a u t o m o r p h i s m of ~(&,V)

£(g,V)

which

fixed is an inner automorphism.

47 ix)

Show by an example

is0

have a unique the form

addition.

that e subring

Such an example over a 2-element

a21 0

0:

all 0

0

~

a32 a32 0 is an automorphism

In fact,

need not

of all matrices

of

show that the mapping

~

a21

0

a31+alla21

a32

but is not additive.

Find another

addition

for

~I~

which

it a ring. x)

State

striction

and prove

xi)

Show

that if

and

Chapter

[4], Jacobson

[1],[2],

2, Sections

([7], Chapter

Mihalev

Gluskin

We will discuss

and Lam

[I], R~dei

ii),

/i], [2],

[1], (~2],

IX, Section [l], Johnson

If], Mihalev [i], Rickart

12), and

[i], [1],[2],

AUTOMORPHISMS groups

in the preceding

we write

all functions dim V >

auto-

Throughout

of functions

on a set

on the right. i,

auEomorphisms

automorphisms

of semilinear

section.

Multiplication

space with

is the group of all linear

Johnson

Martindale

of certain

is the group of all semilinear A

Jacobson

and Steinfeld

notation.

is a fixed vector

/I],[2],

IV, Section

/2],

[5],[6].

only a few properties

is taken to be their composition;

of

see

6), Dieudonn~

[I], Fajans

12),([6], Chapter

OF SEMILINEAR

encountered

gidelheit

[i], R.E.

[1],[2],

[i], Wolfson

we fix the following

(&,V)

Halezov

transformations,

II, Section

ii), ([7], Chepter

III,IV),

3, Section

GROUPS

we have essentially

this section,

IX, Section

[3], Mackey

/i], Morita

1.6

the re-

with all idempotents

and semilinear

([i], Chapters

12), Jodeit

[i], Stephenson

and commutes

([i], Chapter

[2],[6],[7],

[i], Kaplansky

and Satalova

morphisms

on isomorphisms 4), Behrens

Baer

5.14 and 5.15 without

must be a multiplication.

7,8 and Chapter

Y

[3], Rjabuhin

is additive a

[3], ([6], Chapter

IV, Section

Kiokemeister

in 5.17

V, Section

and for related material

of 5.12,

V.

i, then

information

([1], Chapter

Gewirtzman

of

a

dim V >

For further

Gluskin

the ring version

on the dimension

~2,U(&,V),

Baer

of

field.

addition

~

Jail001 Lall 0!]

a31 a32

makes

of a ring with unique

is given by the ring

of

of V

V, (invertible

linear

transformations), M

I

is the group of all multiplications is the group of all semilinear

(m

= (e

,m ) with ? inner automorphisms (i.e.,

W E ~ ),

48 £ = £(A,v). Furthermore,

for

S

a semigroup

(or ring),

let

~(S)

be the group of all automorphisms

~(S)

be the group of all inner automorphisms

where an inner automorphism

of

S, of

S,

of a semigroup or a ring is defined by g -I (x ~ S), as in the case of groups, with the provision that g exists,

must have an identity If set

H

e

element and

g

is a subset of a group

{g E G I gh = hg

for all

must have an inverse

relative

G, then the centralizer

h E H}.

For subsets

H

of

and

H

K

-i = g xg g i.e., S

x¢

to it. in

of

G

is the

G, let

HK = [hk lh E H, k E K]. PROPOSITION.

1.6.1

Further,

A

a normal

subsrou~

is the centralizer

PROOF.

o_f~ ~ ,

~

onto

subgroup

6(£)

o_~f ~

PROOF. For any

with kernel

and of

By 5.8,

c E ~

implies

5.21, M

and

thus

that

~

[

for any

M

and is a normal

M ~ A = im

maps

and maps

~

into

of

~

subgroup

I ~ ~ C(a-)}

onto

of

~,

M.

I

i__~s

= C(M).

~.

~

maps

and by 5.13,

= ~(w,a)(~,b) M

Hence

¢

M

maps

is a normal

~

onto

6(£).

= (ab) -i c (ab) = c¢(w~,ab)

and hence

~

is a homomorphism.

is s normal subgroup of

m

I

$(~).

we have

Hence

7 E g , we have

to show that

onto

= b -I (a-lca)b

(a-lca){(w,b)

~(w,a)~(~,b)

((~,a) E >~), is a homomorphism

I/M ~ $(£).

6(£),

(~,a),(w,b) E ~ , =

I

~ / M ~ ~(~),

is also a normal subgroup of

It remains

i_nn ~

is an isomorphism

~:(w,a) ~ {(w,a)

M,

and

is the kernel of

Since

M

I = MA, a n d

Th__.~emappin $

c~(~,a){(~,b) which

of

~ ~ m -i

Exercise.

1.6.2 THEOREM. of

The mapping

= (¢ _l,m ), it follows I

onto

and the kernel of $(~).

Hence

let

~

~

and

that

M c

restricted

(¢ ,a) E I.

By

~ / M ~ 6(£). I

and to

I.

For any

o E g, x E V, we obtain (ox)(m a) = (7ox)a = ~-l(~g)~(xa) = c[(~x)a] and hence

m a

is linear.

= o~(xa)

= o[[(~e

_l)x]a}

= o[(xmv)a ] = o[x(mva)]

This

together with 5.20 yields

C(~v,a) = ~m a = e m a E a(~). 7 Finally, ~Z~,a~<)

let

~(w,a~,,E

= ~(5~,b)" =

J(~).

Then there exists

But then 5.21 implies

= e _15~ c -i £ J(~) so that V inverse image of ~(£) under ~.

that

(~,a) E I. In

a linear automorphism a = m~b

for some

It follows

particular,

that

I/M ~ ~(£).

~ £ ~ I

b

such

that

and

is the complete

49 1.6.3 COROLLARY. are equivalent:

The following conditions on a semilinear isomorphism

i) ~ E $(&),

ii) ~(~,a) E ~(£),

iii) m a

(~,a)

is linear for some

rE&. PROOF.

Exercise.

1.6.4 COROLLARY.

The identity m a p p i n g i~s ~h___eeonly a u t o m o r ~ h i s m of

leaves fixed every element of PROOF.

Let

automorphism

~

(~,a)

Hence

(w,a) E M

S

PROOF. (~,a)

and

£(g,V)

Sl

S

an___~d S ~

~

and

~(g',V~),

so t h a t

~(w,a)~(i',a ')

these are extensions

of

b

of

Now

~2,U(&,V),

If then

PROOF.

®

0

onto

~

~

~(,

and

,

~

satisfies

Iva = v

for all

is a subgroup of

B

~(w',a~) '

of

say

are isomorphisms of

the c o n d i t i o n s

we m u s t have

of 6.4,

and s i n c e

~ = ~. V

and let

v ~ B].

~,

the m a p p i n g

and

® n A = ~V'

((~,a) E ®)

~ ~

onto

~(&)

are iso-

~ = ~l.

respectively,

6(&), ~

V e r i f i c s t i o n that

m o r p h i s m of

an___dd ~

~

~:(~,a) is on isomorphism of

~

~(w,a) =

~

for some

~ = ~(w,a) = ~£"

For the remainder of this section, we fix a basis

1.6.6 PROPOSITION.

for some semilinear

a = m

Consequently

® = [(w,a) E ~

which

~2,u(V), we have

= ~(g,V)" ~',

£

V*.

are induced by semilinear isomorphisms,

~(~,a) ~

and

o_L

0 = ~(w,a)

be as in 5.12.

~

• (w , ,a l ), respectlvely.

U

By 5.17, we must have

which agree on

By 5.12, both

onto

and t h u s

ba = ab.

and by 6.2, we obtain

Let

onto

Then

For every element

so that

1.6.5 COROLLARY. morphisms of

for some t-subspace

be such an automorphism. by 5.13.

b = b~(~,a) = a-lba E & •

~2,u(V)

®

= ~A,

is a subgroup of

w i t h trivial kernel

~

and that

~

is a homo-

is left as an exercise

(for ~'onto"

part use the function a c o n s t r u c t e d in the proof of 5.15)o For

(~,a) E ~

and extend of

~

define

7

on

V

V, we have that

~

maps

V

onto itself.

~ v =

(vi

E B)

"

°l = 02 = "'" = ~n = 0

Hence

Let Since

Suppose

v ~ = va

for every

s

B

that

maps

v ~ B,

onto a basis

v ~ = 0, where

Then

0 = v~ =

~ ~.(v.a) i= I i l

since the vectors

w h i c h implies that

vla,v2a,...,Vna

are linesrly independent.

v = 0 and 7 is one-to-one. Therefore ~ E A and (w,a) = (~,a~-l)(~&,~). ---l as maps V onto V and is one-to-one; for o E g, x E V, we have (ox)a74

and for

V.

n

~.v. i= l i i

Hence

as follows.

to a linear transformation on

= [(ox)a]7-1 = [(om)(xa)]7-1 = (o~)(xa~-l)

v E B, v ~ = va

(~&,7) E A

implies that

w h i c h proves that

~

= ®A.

v a ~ -I = v.

Thus

(w,a~ -I) E ®

and

50 If

(~,a) E ® N A, then for every

implies that

a = lV

since

v E B, va = v

(w,a) E A

and

B

since

(~,a) E ~, w h i c h

is a basis of

V.

We now discuss a group theoretic construction which will clarify some of the concepts we have studied, 1.6.7 DEFINITION. we w r i t e

~:h ~ ~h'

see Hall

Let

H

The set

and

We say that a group

G

of

The mappings

H

and

identify

and

with

1.6.8 THEOREM. then if and

G G

If

is a group, A

A (determined by

and

PROOF.

Let

G

H

with

H

it is convenient

to

and we do so henceforth. H

with a group

G, G = HA, H ~ A = i.

and a normal subgroup product o f

be a semidirect product of

(more precisely

are easily seen to be isomorphisms

is a semidirect product of a group

is a semidirect

A

to a group c o n s t r u c t e d as

into the semidirect product;

is a normal subgroup o f

G

~,).

is isomorphic

a ~ (l,a)

is a group having ! subgroup H N A = i, then

G

their respective images, G

be a homomorphism,

(hh~,(a~h~)al)

~") if

h ~ (h,l)

A

~:H ~ ~(A)

is a semidirect product of

A, respectively, H

be groups and

together with the m u l t i p l i c a t i o n

H with

"determined by a h o m o m o r p h i s m above.

A

HX A

(h,a)(h~,a ~) =

is a s e m i d i r e c t product of

([i], Chapter 6, Section 5).

H

H

with

with

A.

A

A,

Conversely,

such that

G = HA

A. Then

I(h,a)(h',a')](h',a" ) = (hh~(a~h,)a')(h~,a") I

= (hh'h",[(a,~h,)a

ii

t

l~h,,a ) = (hh'hH,(a~h,~h,,)(a ~hH)a ~)

= (hh'h",(a~h,h,,)(a'~h,,)a")

= (h,a)(h'h~,(a'~hH)S")

= (h,a)[(h',a')(hH,a")] and

the multiplication

is associative. (h,a)(1,1)

Further

= (hl,(a~l)l)

= (h,a),

(l,l)(h,a) = (lh,(l~h)a) = (h,a), so that

(i,i)

is the identity of

(h-l,a-l~

_l)(h,a)

G.

Finally,

= (hh-l,(a-l~

_l)~h a) = (l,a-la)

h w h i c h shows that

= (i,i)

h

(h -1, a -13h_l )

is a left inverse of

(h,a).

It follows that

G

is a group. Now we note that (h'a)-l(l'b)(h'a) = (h~l'a -i ~ -i )(h,(b~h)a h =

so

that

A

is a normal

subgroup

(l,(a-l~h_l~h)(b~h)a) of

G.

It

is clear

) = (l,a-l(b~h)a), that

G = HA

and

H ~ A = i.

51 Conversely, such that

let

G = HA

the form

g = ha

implies that g~ = h~a~;

G

and

be a group with a subgroup H N A = i.

with

Since

h E H, a E A.

H

and a normal subgroup

G = HA, any

g E G

can be w r i t t e n in

This r e p r e s e n t a t i o n is unique since

h~-lh = a~a -I ~ H N A = i, so

h = h ~, a = a j.

A

Next let

ha = h~a ~

g = ha,

then gg~ = (ha)(h~a ~) = hh~(h~-lah~)a~,

where

h~-lah ~ E A

a~h = h~lah

since

A

is a normal subgroup of

(a E A), we see that

~h E G(A).

(a~h)~ht = h ~ - l ( h - l a h ) h ~ so the m a p p i n g then can write

~ : H ~ ~(A)

(i)

For

G.

For

h E H, w r i t i n g

h,h ~ E H, we have

(hh~)-la(hh ') = a~hh~

defined by

~:h ~ ~h

(h E H)

is a homomorphism.

We

(i) as gg~ = (hh~)[(a~h~)a~ ]

or, w r i t i n g the elements of (2),

G

(2)

as ordered pairs in

H X A,

(i) becomes by virtue of

(h,a)(hJ,a I) = (hh~,(a~h~)a~). Therefore

G

is a semidirect product of

With the n o t a t i o n of 6.8, are equivalent

let

to saying that

follows easily that

B ~ H.

H; in fact, G

H

G/A = B.

H

(one also says that the extension

Let

G

ii)

~

maps

Every element of

splits over

H

with

A).

A

by

For a discussion of group

H

with

A

determined

A.

G(A).

commutes with every element of

~ $(h)

is a semidirect product o f with

A.

~

®~ = ~ G I. ~' N I

C(&)

with

A

is a normal subgroup of ®

with

A.

is also a semidirect product of

and

I

is

is a normal subgroup of

~,

so that by

A g a i n by 6.6 we have ~(h)

with

It is left as an exercise to show that

= iV, A

A

A.

is a semidirect product of

and hence

and

H

We have seen in 6.1 that

6.6 and 6.8, ~

I = ~JA,

It

is an extension of

Exercise.

a semidirect product o f

Now let

G

onto the identity o f

1.6.10 PROPOSITION.

® ~ 6(A)

exactly once.

Then the followin$ statements are equivalent.

is the direct product o f

PROOF.

A

be a semidirect product o f

~:H - 6(A).

G

PROOF.

G = HA, H n A = i

([i], § 50).

i)

iii)

The conditions

is a very special extension called a split extension of

1.6.9 PROPOSITION.

H

A.

intersects each coset of

by

by the h o m o m o r p h i s m

with

In terms of (group) extensions, G

A

extensions see R~dei

H

A.

®' ~ $(h),

I, so that the proof of the

second statement of the proposition follows along the same lines as the proof of the first.

52 1.6.11 REMARK. product o f

G(&)

We can make 6.10 more precise b y w r i t i n $ the semidirect

with

A

(and thus also of

~(&)

with

pairs a n d exhibiting the determinin$ h o m o m o r p h i s m account th___~eisomorphism in 6.6 between (~,a) E ~

as the product

of the h o m o m o r p h i s m Let

~

~ E G(&)

(m,b) E ~

For any

and

(~,a~-l)(s~,~)

~.

To this end, we take into

G(~),

th__~erepresentation o f

in the proof o f 6.6, and the construction

in the proof of 6.8.

and define

b

vb = i=l(~i~)vi ~ Then

~

A) in terms of ordered

on

V

by

n v = i~=l~iVi

if

and we obtain the function

v E B, we have

va =

o.v. i=l 1 1

(v E B).

M :a ~ b-lab

for some

oi E &

n

(a E A). and

Let

vi E B

a E A.

whence

n

v(b-lab) = (va)b = (~lOiVi)bi = = (oiw)v i •= i=l since

b

leaves the elements of

B

fixed.

=

~ :a ~ a~

(a E A)

where

~

n

va

Hence

~ (~.~)v. i=l z l

if

va =

O.v. i=izl

(v E V, v i ~ B).

(3)

Thus the semidirect product m u l t i p l i c a t i o n is given by i

(~,a)(~',a') where

aw

= (~x~',a w • a ' )

(~,~'

then for any

v E B

we have

can be represented as a

for

a , a ' ~ A),

i s g i v e n by ( 3 ) .

We are now able to express automorphisms of

c

e G(A),

i < i < n

vc = B ×B

and every

~ o.v. i=l z i

0

in matrix form.

for some

m a t r i x over

v E B, and

~

A

c~i E f~

having

otherwise.

and

oi

If

c E £,

v. E B. z

in its (v,vi)-entry

This m a t r i x

C = [~ij]

row finite (i.e., every row has at most a finite number of nonzero entries). versely,

it is clear that to every row finite

uniquely an e n d o m o r p h i s m of

(A,V).

B× B

matrix

C

is an isomorphism of

£

A, where

are added and m u l t i p l i e d as usual

Now let

c E ~

c~(®,a) where

onto the set of all row finite

B ×B ("row by

is possible because of row finiteness).

and

(0~,a) E ~ .

Then, as we have seen above,

= a -ica = 7 -l[(a~ -l)-Ic(a~-l)j~

~ = (~A,-a) E A, a~ -I = (~,a~ -I ) E ®.

for some

Con-

It is easy to show that the c o r r e s p o n d e n c e

c ~ C = [~ij]

column" m u l t i p l i c a t i o n

is

one can associate

matrices over

the matrices

Hence

o i E A, v i E B, and thus

For

v E B, we have

(4)

vc =

~ q.v. i=l z t

53

v(a~-l)-ic(a~-~

---l aa E ~

using the fact that 7

to

A = [~ij]

n = ( -~=l oli v ii) a ~

= (vc)a~-i

in matrix

twice.

Then if

c

=

n ~ (a,~)v. i= I ~ l

corresponds

to

C =

[Wij]

and

form, we obtain by (4), -1

[Wij]~(w,a)

= A

We have thus proved,

using 5.13,

the following

if the corresponding

endomorphism

For the finite dimensional some time,

see Jacobson

to Halezov

[I] and Gluskin

1.6.12 THEOREM. elements in

of

in

£

vector

([2], Chapter

Let

~ = £(A,V)

£, and for

[Wij~]A. Call a matrix

C

invertible

spaces 2,

the next result has been around

Section 5);

the semigroup version

for

is due

[31. (g,V)

be a vector

in matrix

[~ij] E £,

result.

is invertible.

form.

space with

Let

m E 6(&),

dim V > A

i, and write

be an invertible

the

matrix

let

[~ij]~ = [Vij~] , [Vij]¢ A = A-I[wij]A. Then

~(~,A) = S e A

is an automorphism

is of this form.

Moreover,

of

~, and conversely,

~(w,a) 6 ~(~(~,V))

We see that every automorphism

of

~

every automorphism

i f and only if

form

~

and an inner automorphism.

From the above discussion

~

is a semidirect

C(A) with

matrices

over

A

product of

w C ~(A).

is the product of an automorphism

that

of

of the

and 6.10, we see

the group of invertible

BX B

with multiplication i (w,A)(~',A')

where

for

=

(tlJoJ/,A w

"A')

A = [~ij], A ~ = [~ij~].

1.6.13 EXERCISES. i)

Define a semidirect

a homomorphism nition

of

H

product of a semigroup

into the endomorphism

is to be the same as for groups.

semigroup of semilinear is a subspace

of

transformations

V; let

semigroup

Let on

(&,V) V.

rank (w,a) = dimVa.

Ca(V ) = [(~,a) E ~(V) I r a n k a Find all ideals of ga(V)

~(V),

show that

and that each maximal

with a maximal

subgroup of

subgroup g(V)

P (V) Q

of

H

with a semigroup of

A, otherwise

(as groups).

by using

the defi-

he a vector space, ~(V)

be the

(~,a) E ~(V),

Va

For

show that

Define < Q}.

is a semidirect ~(V)

A

product of

is a semidirect

~(~)

product of

with G(A)

54

ii)

Show that

~2(V ) ~ ~ O ( I v , , G ,IV;P),

where

G

is the semidirect product of

_

~(&)

with

_

~

(as groups) determined by

Pgi = (~&'(vD'ui))" iii)

~(s,z) for

Let

(Hint: Consider

(g,V)

defined

by

x~(a,z ) = xa +z

translation.

Let

+ T ~ V

a)

For

s E f~

( x E V)

is

is called a homothetic

A, H, and

homothetic transformations,

T

and

~

~lli~&_ and

called

z 6 V, the function

an sffine

transformation;

transformation;

a(ml,z)

is called a

be the groups of all affine transformations,

and translations,

(the additive group of

A N T = iV, so that

with

xli,(w,~),k ) = (x,ui)w~vx.)

be a vector space.

~ E g , ~(m ,z)

=

~:C(A) ~ 6([i~& )

respectively.

Show that

is s normal subgroup of A, A = AT, + is a semidirect product of A with V ; express this

A

V), T

explicitly. M ~ ~&

b)

, T

is s normal subgroup of

a semidirect product of iv)

~-

H

is

(automorphism) groups related to vector spaces,

see

with

Find the center of the following groups;

v)

Use this to strengthen the assertion of 6.5.

For information on various Baer ([i], Chapter VI), Dieudonn~ 1.7 1.7.1 DEFINITION. is an ideal of ii)

the property

o

[4], Gluskin

[2],14],[6],

~lotkin

([i], Chapter IV).

EXTENSIONS OF SEMIGROUPS AND RINGS

i) A semigroup

K

is an extension of a semigroup

S

if

S

K.

A congruence

~:S ~ S I

if

~,A,I,®,A,H.

Show that "element ~' in the statement of 6.4 can be substituted by

"idempotent".

only if

H, H = MT, M fJ T = ~V' so that

V+; express this explicitly.

that

~

on a semigroup

a ~ b

implies

is a homomorphism, ~p = y~

is an equivalence relation on

ac o bc, ca o cb

then the relation

is a congruence,

is a congruence on

S

o

for all defined on

a,b,c ~ S. S

by:

called the congruence induced by

S, then the set

S/~

of all classes of

with

If

x o Y ~.

o

S

if and

Conversely, can be given

s m u l t i p l i c a t i o n in a natural way. iii)

An extension

only congruence on

K

dense extension

of

containing iv) for all

K

K

K

of

S

is dense if the equality relation on

whose restriction to S

is maximal

S

K

is the

is the equality relation on

if there exists no dense extension

KI

S.

A

of

as a proper subsemigroup.

A semigroup

S

is weakly reductive

x E S, implies that

a = b.

if for any

a,b E S, ax = bx, xa = xb

S

55 The m a i n result of this section is the statement that dense extension of each of its nonzero ideals £u(V)).

1.7.2 DEFINITION.

w r i t t e n on the right,

and

Let

S

be a semigroup.

x(Xy) = (x0)y

%

The set I

(X,O)(%~,O)

~(S)

I

= (~h ,PO )

ha

and

pa

defined by:

sometimes write

~ = (~,O) E ~(S)

1.7.3 LEM~iA.

I

and

~(S)

PROOF.

Let

= ~wa'

S,

The pair

S.

For

XPa = xa

under the I

) = (xp)01

a E S, the

(x E S)

a, respectively;

xw = xp, wx = ~x

(x E S).

are called m s = (Xa,Pa) We will

(a E S, w E ~ ( g ) )

~(S).

Exercise. Let

K

be an e x t e n s i o n of a semigroup

and

p

k

are functions on xkx = kx,

T

S

I

I a ~ S}.

= T(K:S):k ~ ( k , p k )

Then

(x,y E S).

~(S) = ~ a

~a w = ~aw

is an ideal of

1.7.4 LEMMA.

k

on

is a right translation,

= ~(~ x), x(pp

of

p

With the n o t a t i o n as in 7.2, we have

~a so that

with

P

(X~)x ~(S)

%a x = ax

a.

S, w r i t t e n on

of b i t r a n s l a t i o n s of

where

inner left and inner right translations induced by is the inner b i t r a n s l a t i o n induced by

on

(xy)p = x(yp)

(x E S), is a s e m i g r o u p , t h e translational hull functions

~

(x,y E S); a function

is a left translation,

(x,y ~ S). I

multiplication

A function

~(xy) = (~x)y

is a right translation if

is s b i t r a n s l a t i o n if

where

is a m a x i m a l

For this we need some preparation.

the left, is a left translation if

(X,p)

gu(V)

(and the c o r r e s p o n d i n g statement for

K

and

~

(k,pk)

E ~(S)

(k E K)

(x E S).

into

~:a ~ ~a

Let

defined by

xp k = x k

is a h o m o m o r p h i s m o f

= TIS , we have

S

S.

~(S)

a n d maps

S

onto

~(S).

is one-to-one if and only i f

S

Letting

is weakly

reductive. PROOF.

That

from the a s s o c i a t i v i t y in xa = xb

(x E S)

K.

and that

For

a,b E S,

which implies that

~

f

is a h o m o m o r p h i s m follows immediately

~a

=

~h

is one-to-one

if and only if

ax

=

bx~

if and only if

S

is weakly

S

The next

reductive. Hence for a w e a k l y reductive

S, we may identify

result is due to Grillet and Petrieh 1.7.5 THEOREM. let

~ = T(K:S).

Let

Then

K K

with

be an extension of a weakly reductive semigroup is a dense extension of

S

if and only i f

one-to-on____ee, and is a maximal dense extension if and onl~ i f ~(S).

If we identify

S

w i t h all subsemigroups o f ~(S)

•

~(S).

[i].

with ~(S)

~(S),

~

then dense extensions o f

containing

~(S)

T

also maps S

S

and

is K

onto

may be identified

and maximal dense extensions with

56 PROOF.

Let

congruence then

for

on any

T.

o

~a

Let ~(S).

if

K

already K

K

so

that

dense

proved

that

K\~(S).

and

that

of

p

of

ax = bx,

of

~(S)

o

be a

a ~ b,

×a = xb. in the c o n g r u e n c e

that

K

is a d e n s e

containing

w o i,

then

the h y p o t h e s i s , aw = aw I

for

is the e q u a l i t y

S

K

is not

that

of

S,

~(S)

~(S) of

for w h i c h

see

that

for

~(S) any

~

whose E ~(S),

a

we o b t a i n

all

a ~ S

relation,

K~

~(S)

since

so

then

is a p r o p e r

is a d e n s e

a maximal •

dense must

is a m a x i m a l

is a d e n s e ~(S)

T

S

is

is a d e n s e

containing

extension

be onto.

dense

extension

of

~(S)

subsemigroup

extension of

S

S.

Hence

To p r o v e

extension

of

of

of

the ~(S).

~(S).

as a p r o p e r

subsemigroup,

a a~x'

on

S

~xP

extension

= ~x a

~ k x = a~x'

by

and w r i E i n g

T = T(K:~(S))

a dense

has

of

~

(x E S). ~Xp

=

= ~xa

(k,p)

the p r o p e r t y

~(S).

(x E S).

that

ka

aT = ~T

Therefore

~(S)

It is e a s y

k ~,

=

with

p

a

P

a ~ w

is a m a x i m a l

w

to

•

so that

dense

K

extension

~(S).

For

S

a semigroup,

in P e t r i c h

[3],

1.7.6 maximal its

dense

ideals

containing Let

extension

of

S,

of

T

since

a,b E T

S

the e q u a l i t y K

T

S

2

appears S

= {ab I a,b ~ S}. here

for

the

be a s e m i g r o u p

of

S.

Then

K

The n e x t

first

such

it f o l l o w s

D.

relation,

is a d e n s e

is m e n t i o n e d

time.

that

2

S

is a m a x i m a l

= S

dense

of

that

K

K

containing

is a d e n s e

S.

and

let

extension

of

D

If then

so g

bd E

T

and

is a c o n g r u e n c e o

extension

restricted of

S.

But

S

then

D

K

K

of

K

be a

of e a c h

of

Let

then

obtain

whose

is a d e n s e

T.

s 6 S,

is an ideal

on

to

Since

extension

containing K as a s u b s e m i g r o u p . If 2 since S = S. H e n c e for any d ~ D, we

of

theorem

S. be an ideal

is an ideal

is an ideal

since

Let

extension

extension some

let

its p r o o f

THEOREM.

PROOF.

S

and

we

=

(~,p) ~ ~ ( S ) ,

Consequently is not

let If

Then ~x

check

and

is c o n t a i n e d

If

7.3 and

~

extension

to show

~a

k

and

KT,

be an e x t e n s i o n

a E

Define

theorem relation.

we c o n c l u d e

T

relation.

wa = w l a

with

it s u f f i c e s

let

o

is o n e - t o - o n e ,

so by

extension

K

is a m a x i m a l

Let and

that

on a s u b s e m i g r o u p

w = i

be a d e n s e

containing

have

the

so by h y p o t h e s i s proves

the e q u a l i t y

Thus

of

is the e q u a l i t y

is o n e - t o - o n e .

Hence

i.

S

~(S).

K

converse, We

is

maw

=

Identifying

properly

T

TIS

' ~a ~ o ~a ~l,

reductive. of

that

if

~(S)

~ W~a ~aw

extension

to

xa o xb,

a congruence

to

~a

= ~a'

weakly

Recalling

be

restriction we h a v e

be as in the s t a t e m e n t

ax ~ bx,

if and o n l y

Let

K

restriction

h a = ~b , p a = p b , w h i c h

by

extension

and

whose

x E S,

Consequently induced

S

K

of

T.

to

for ~ S

Consequently

restriction

restricted

be a d e n s e

sd = a(bd)

is the e q u a l i t y g

D

s = ab

to

relation T

is

S on the

is K

57 equality

relation on

T, and since

D

is a dense extension of

is the equality relation on all of which by maximality extension

of

of

K

that

Thus D = K.

For a weakly

reductive

1.7.8 LEMMA.

K

is a maximal

dense

semisroup

S

for which

S 2 = S, ~(S)

N(S).

Let

0 ~ (k,p) E ~(S).

S = ~°(I,G,A;P)

[i].

be s Rees matrix

Then there exist a subset

~:A ~ G

(written on the left) and a subset

~:F ~ G

(written on the risht) ~(i,a,~)

A F

o_~f I of

A

semisroup

and let

and functions and functions

~:A ~ I, ~:F ~ A,

satisfying

(~i,(~i)a,~)

if

i E A

= 0

otherwise

(i,a(~),D~)

if

D E r

(i,a,B)p = 0

iii)

Therefore

that S

Exercise.

The next lemma can be found in Petrich

ii)

T, it follows

is a dense extension of

dense extension of each of its ideals containin$

PROOF.

i)

D

T.

1.7.7 COROLLARY. is a maximal

implies

D.

otherwise

i E A, PD(~i) # 0

if and only if

~ E [~, p(D~) i # 0

an_d i_~f s_~o, then

PD(~i)(q°i) = ( ~ ) p ( D ~ ) i , for all

(i,a,~) E S \ 0 .

PROOF. k, Q

The condition

is nonzero.

there exists # 0

implies If

means

But if, e.g., Xx # 0

y E S

such that

then since

y(kx) # 0

p # 0, and conversely,

that at least one of the functions

so that

S

is a Rees matrix semigroup,

(yp)x # 0

# O, then there exists

j 6 1

k(i,a,D)

-l = k[(i,a,D)(j,pDj,D) ] =

-1 [k(i,a,D)](j,pDj,D)

that

X(i,s,~)

Suppose next that is the identity

of

is of the form

l(i,l,D) G.

= (j,b,D) # 0

Then for some

(j,b,~) = ( j , b , ~ ) ( m , p ~ , ~ )

=

= k[(i,l,D)(m,p~,~)] Consequently

j = k

of

depends

k(i,l,~)

and

b = c

only upon

and

and hence both functions

k(i,a,D)

which implies

i

(l,p) # 0

such that

yp # 0.

are nonzero.

p~j # 0

and

# 0

( , ,~). and

k(i,l,~)

m ~ I, we have

= (k,c,~) # O, where

p~nn # 0

and thus

[X(i,l,~)](m,p~,~) = k(i,l,v)

= (k,c,~) # O.

which shows that the value of the first i.

Thus

two indices

58 This makes

it possible

~:A - I, ~ : A ~ G If now

to define

by the formula

k(i,a,D)

A = [i ~ I I k(i,l,D)

k(i,l,z)

= (~i,~i,v)

# 0, then for some

j E I, pDj # 0

= (~i,~i,p)(j,p~ja,p)= with

i ~ A.

The same calculation

h(i,l,D)

= 0

so that

argument

establishes

For any

i ~ A.

and the functions

~ 0]

(i E A). so that

(~i,(~i)a,D)

shows

that if

This proves

k(i,a,H ) = 0, then we must have

part i) of the lemma;

a symmetric

part ii).

i,j E I

and

D,v 6 A, we obtain

(~i

1 (j,I,D) [%(i,i,~)]

if

i E A

= ~(J'

L (j,l,~)0

otherwise

= {(j,pg(~i)(~i),~ )

if

i ~ A, P~(~i)

# 0

(1) 0

otherwise

and analogously [(j,l,H)p](i,l,m)

Since by hypothesis,

For the remainder

1.7.9 THEOREM.

For

as in 7.8 and define

I

Then

a E ~u(V),

b

{

let

is essentially S = ~2,u(V) a

~(B~)vH~

(2) are equal,

the assertions

taken

be a pair of dual vector

from petrich

[2].

an___~d 0 # (k,p) ~ ~(S),

an___dd b if

(U,h,V)

let

A,~,~,F,~,~

b_!e

by

~ E F

( T v E V),

[0

=

(2)

follow.

functions

(VvH)a =

~ 0

otherwise

of this section,

The next result

b(uiT)

~ E F, P ( ~ ) i

the left hand sides of (I) and

in part iii) of the lemma

spaces.

,(D~)p(D~)i,m)

= {(J 0

otherwise

u i(~i)T

if

0

otherwise

is the ad~oint kr = sr,

i E A

of

rp = ra

a

(uiT E u). in

U, and

(r ~ ~2,u(V)).

(3)

59 PROOF.

For any

~vp ~ V

and

u.rm ~ U, using 7.8 we obtain

((~vg)a'ui~) = { (~(p,)v ~,ui~ )

if

(O,ui,~)

g ~ F

otherwise

=~(~)(v ~,ui)~ if p E T 0

otherwise

=S~(v

[

,u i)(~i)m

if

i E A

0

otherwise

(~v ,u i(~i)T )

if

(~v ,0)

otherwise

i C A

= (~v , b ( u i T ) ) hence

b

is an adjoint of

unique sdjoint of

s

For any

E V

?v

in and

a

so that

a E £u(V)

and 1.2 implies that

U. [j,~,~] E ~2,u(V), using 7.8 we obtain

~vg(~[j,o,v]) =I~vp[~J'(~J)°'~]

if

j E A

0

otherwise

~(v ,u .)(~j)~v

if

0

otherwise

S~(~)(vD~,uj)~v v 0

if

j E A

D ~ F

otherwise

=I ~(D~)v ~[j,~,v] 0

if ~ E F otherwise

= VvD(a[j,o,~]) which proves the first part of (3).

~vB([j,~,~]p ) =

~

Finally

v [j,o(~,),v~]

L 0

~

if

~ E F

otherwise (v ,uj)o(v~)v

=L 0

if

v ~ F

otherwise

= ~(vp,uj)ov a = VVD([j,~,~]a ) which proves the second part of (3).

60 The following result is new; 1.7.10 THEOREM.

gu(V)

for

g(V)

it is due to Gluskin

[5].

is a maximal dense extension of each of its nonzero

ideals. PROOF. of

By 5.11 we know that

K = gu(V).

Since

weakly reductive and

S

S = ~2,u(V)

S 2 = S.

it suffices to show that

and thus T

for all

r E S

(k,p) E ~(S) If

a,b E K

S

is

according to 7.5 and 7.6,

is one-to-one and maps

with the n o t a t i o n of 7.9.

(a - b ) r = 0

it follows easily that

Thus to prove the theorem,

T = ~(K:S)

onto part follows from 7.9 since for a given (~a,pa) = (~,p)

is contained in every nonzero ideal

is a Rees matrix semigroup,

K

~(S).

The

we obtain

and

which by 3.3 implies

onto

a~ = bT, that

then

a = b

ar = br

and thus

is one-to-one. 1.7.11 COROLLARY. PROOF.

For every nonzero ideal

I

gu(V), we have

~(I) ~ gu(V).

This follows directly from 7.10 and 7.5.

We will now obtain the corresponding results £u(V).

of

for nonzero

For this we need the concepts and statements

of those we have seen for semigroups.

(ring) ideals of

for rings which are analogues

To this end we substitute semigroups with

rings w i t h the following modifications. In 7.1:

i) the definition remains

speak of ideals; only ideal of

iii) an e x t e n s i o n

K

K

the same; of a ring

whose intersection with

S

is

ii) instead of congruences, 0

we

S

is essential if

is the

0

(such an ideal is called larse);

iv) is expressed by saying that the annihilator ~(S) = ~a E S l as = sa = 0 is

for all

s C Sj

0. In 7.2 we only add that every left and every right translation also be an

additive homomorphism. In 7.6 the condition (i.e.,

S2 = S

can be w e a k e n e d to

the same kind of condition with

With these modifications,

S

2

defined as in ring theory).

all statements

of this assertion is left as an exercise

n [ ~_isiti~_ I si,ti ~ S} = S

in 7.3-7.7 remain valid;

the proof

(work with ideals instead of congruences

as is usual in ring theory). 1.7.12 LEMMA.

~u(V)

PROOF.

is a nonzero ideal of

~u(V),

If

I

is contained in every nonzero ideal of

so that by 5.11 we must have

£u(V),

~2,u(V) ~ I.

The following result appears to be new.

then

[[iI

£u(V).

is a nonzero ideal of

But then by 2.6 we obtain

61 1.7.13 THEOREM.

£u(V)

is a maximal

essential

extension of each of its nonzero

ideals. PROOF.

We know by 3.6 that

(ring and semigroup) ideal of theorem 3

~ = £u(V). it suffices

is regular,

that

condition

~

we have

essential

follows

p

= p132 .

that

32

maps

32

(~,pl)

essential

~

of

~

of 7.5,

onto

is a homomorphism.

to prove the

extension

5, it suffices

isomorphism

the

in every nonzero

of 7.6, in order

is a maximal

extension of

as in 7.4 is a (ring)

of

3.

in order

Since

to prove

to show that

~(3).

By the ring

That the kernel of

T

is

from 3.3.

To prove that I

£

is contained

~(~) = 0, so by the ring analogue

analogue of 7.4, we know that trivial

is regular and hence satisfies

Hence by the ring analogue to show that

is a maximal

• :~ ~ ~(3)

3 = 3u(V)

32 = 3; by 7.12, 3

T

maps

Then for any

into itself;

onto

r E 32,

is a Rees matrix

E ~(32).

~

semigroup).

Thus

p~

maps

(X,p) E ~(3). e 6 32

rp ~ = ra

Next let

such that

32

into itself.

X~ = X132,

r = re

hlr = h(re) = (hr)e E 32

a E ~

there exists

~Ir = ar,

let

there exists

similarly

By 7.9,

~($),

It follows

(recall

so that

~

that

such that

(r E 32).

n

If

c E 5, then

c = k~=irk

for some

n

n

and similarly

Cp = ca. that

T

1.7.14 COROLLARY. PROOF.

Since

maps

then implies

n

Xc = h ( ~k=l rl) ~Xrk ~ = k=l

which proves

r k E ~2' which

a E £,

£

n ac

= k=l ~ X'rk = ~ =lark = ak~=irk =

onto

it follows

For every nonzero

This follows directly

that

T

maps

a

onto

(X,p),

~(3). ideal

1

of

~u(V),

we have

from 7.13 and the ring analogue

~(I) ~ ~u(V).

of 7.5.

1.7.15 EXERCISES. i)

Show that every regular

semigroup

is weakly

reductive.

Deduce

that a

regular ring has trivial a n n i h i l a t o r ii) that

~

reductive

Let

~

be an isomorphism

induces

an isomorphism

and we identify

to a homomorphism

of

(~,p)~ = (~,p), where iii)

Let

respectively. Now let

S

S

with

~(S) ~x =

into

(X,p) ~ ~(S)

Show that the function

of

~

be invertible

onto a semigroup ~(S~). ~

If

S

S ~.

Prove

is weakly

is the unique extension

Do the same for rings.

of

(Hint: Let

xp = [(x~-l)p]~.)

~, p p

S

onto

show that

~(S~)o

and and

~(S)

~(S),

[~(x~-l)]~,

be a semigroup We say that

of a semigroup

~

be a left and right

are permutable and suppose

that

if X

translation

(Xx)p = k(xp) and

p

of

S,

(x ~ S).

are permutable.

62

6(X,p) is an automorphism

of

S.

(k,p),(~t,0 I) E ~(S), S

with

~(S),

morphism

of

iv)

that

Let

~

and

p

v)

Let

~

x(Xy) = (xp)y

Deduce

6(X,p) = e(x,p)l S

induced by

(X,p).

show that for any

In such a case~

where

s(X,p)

if we identify

is the inner auto-

Do the same for rings.

We call

be any ring and

(k,p),(k',p ~) E ~(~).

Deduce

that if

~(~) = 0

6(k,p )

Show

that for every

and

(X,p),(X~,pt)

a

E ~(~),

are permutable. be any ring, ~

(x,y E ~),

and

Prove

p

be functions

that for any

on

~

such that

r,s E ~,

k(r+s)

- (kr+ks)

E ~r(~),

b)

(rs)p - r(sp) E Z~(~),

(r+s)p

- (rp+sp)

~ ~(~).

that if

~r(~)

= ~£(~)

it suffices

e.g., when

= 0, for a pair

that for all

~

is a subring

~

define

(k,O)

of functions

x,y E ~, x(ky) = (xp)y. of

£u(V)

containing

on

~

Show

Su(V)

to be a

that this is

for some dual

(U,V). vi)

In any ring

xoy that under

vii) Define

Let

the circle

a

and

a mapping

a~

~(a,at )

Find an invertible

and

morphisms, viii)

Q(~)

be elements

~

~ a

of a ring

by

~

with

identity.

such that

a o a

~

= a' o a

= 0.

(X ~ ~).

oxoa (X,p) ~ ~(~)

the automorphism

prove

for which

~(a,a~)

is called

with

X

and

p

are permutable

quasi-inner.

and all generalized

Denoting

inner auto-

~(~) c ~(~) ~ Q(~) ~ ~(~).

~ = ~U(&,V)

a)

~(~) = ~ b l ~

b = ~V+a,

b)

Q(~) = is bl~

b ~ ~U(&,V)

c)

If

dim V >

i, prove

a ~ ~, b and

is finite dimensional,

d)

If

V

is infinite

e)

If

g

is a field with more

than one element.

is a semigroup

the sets of all quasi-inner

respectively,

o

by

For

V

composition (x,y E ~).

composition

bitzanslation

~(a,a I) = ~(k,p); ~(~)

the circle

= x+y+xy

~(a,a,):x

and

reductive,

are permutable.

X(rs) - (kr)s E ~r(~),

the case,

by

is weakly

pl

a)

bitranslation

Show

S

(x E S)

inner automorphism.

~

pair

If and

(Xr)pl -X(r0 I) E ~(~).

r E R,

then

prove

~(S)

seneralized

k

:x - (x-lx)p

b

statements.

is invertible].

then

dimensional,

the following

invertible].

then

~ @ $(~) = ~(~) = Q(~). ~ = ~(~) C ~(~) C Q($).

than two elements,

then

~(~)

has more

63 f) ix) onto

Q($) = ~(~)

Let

B ~.

B

and

if and only if

B~

Prove that

~

O(&) = 6(g).

be B o o l e a n rings and

~

be a one-to-one

is a ring isomorphism if and only if

of the corresponding semigroups of

B

and

B~

function of

~

B

is an isomorphism

under the circle composition.

Give

an example of an isomorphism of the additive groups of two Boolean rings which is not multiplicative. x)

Prove that if

~

is a Boolean ring,

Boolean ring has trivial annihilator,

so

then

~(~)

is also.

~ ~ ~(~) ~ ~(~)

of a Boolean ring into a Boolean ring with identity.

Show that a

provides an embedding

Do the same for a ring without

proper zero divisors. xi)

Prove that every extension of a Boolean ring by a Boolean ring is again

Boolean. xii)

Let

~

be a commutative ring without proper zero divisors.

the field of quotients of

3,

embedding of

Show that

xiii)

~

into

Let a ring

zero divisors

~. ~

and

~

be the image of

~

in

~

Let

~

be

in the usual

~(~) ~ i~(~l).

be an extension of a ring

if and only if

~

~.

Show that

~

has no proper

has no proper zero divisors and the extension is

essential. xiv)

Let

Prove that

~ ~

be a ring with

~(~) = 0

and

~

has unique addition if and only if

be an essential extension of ~

has unique addition,

the identity a u t o m o r p h i s m is the only endomorphism of

~

~.

and that

leaving every element of

fixed. xv)

A semigroup

(essential)

(or ring)

S

is said to be uniform if

extension of each of its nonzero ideals

S

is a maximal dense

(cf. 7.10 and 7.13).

Show

that the following semigroups and rings are uniform:

xvi)

a)

every simple semigroup or ring wlth identity,

b)

~x I0 < x < i}

under multiplication,

c)

ix I0 < x < i}

under multiplication,

d)

nonnegative

integers under addition,

e)

the semigroup of all transformations on a nonempty set,

f)

the ring of integers,

g)

the ring of polynomials

Let

R

be the field and topological space of real numbers, R[x]

ring of polynomials over functions m a p p i n g

R

set of all fractions zeros, and

in one indeterminate over a field.

R

in the indeterminate

into itself. p(x)/q(x)

deg q(x) - deg p(x) ~

maximal essential extension of

A . n

p(x),q(x) E R[x],

Show that

be the

be the ring of continuous

For every nonnegative integer n, let

such that n.

x, C

ic(An) = A °

q(x)

An

be the

has no real

and that

A°

is a

64

For the material

concerning

see Grillet

and Petrich

in Petrich

([5]j Chapter

Petrich

[i],[2],

B.E. Johnson

R~dei

ideal extensions

[i], Petrich IIl);

transformations

Lambek

([i], Chapter

related

article

a systematic

see Everett

([1],§§52,53,54,85);

[i] and the survey

linear

[i],[4],

in rings,

Petrich

12].

13].

subject,

in semigroups,

discussion

[i], MacLane

(c) in both semigroups

to the present

IV), Petrich

and translations

and

and rings~

see

For semigroups consult

can be found

[i], M~ller

and rings

Gluskin

[4],[5],

of

65

PART II

SEMIPRIM_E RINGS W I T H M I N I M A L This

is a study

of the class

restrictions

such as simplicity,

properties.

Prime

rings with

one of these being containing one-sided provided

that

a nonzero ideals

of finite

transformation

results

rank on an a r b i t r a r y

concordantly.

prime

rings w h i c h

are essential

in several

ways.

of semigroup

socles

Isomorphisms

of Rees m a t r i x

Each properties class.

of the classes

of rings

of the m u l t i p l i c a t i v e

This

is possible

are

all

with

a nonzero

the rings linear

of this

socle.

isomorphic

transformation

in several ways terpreted

in 1.2.

After

of n e e d e d

specified

II.l.l NOTATION. the set of all elements a ~ A, b E B.

If

of the entire

rings.

have

belonging

unique

by the

to the given

addition.

several

characterizations says

that

transformations

latter

itself

characterization

rings

they are p r e c i s e l y

containing

class was

of the class

of prime

a nonzero

characterized

can thus be in-

of concrete

rings

appearing

characterization.

an i n t r o d u c t i o n

ring unless

This

characterization

all the c h a r a c t e r i z a t i o n s

arbitrary

charac-

isomorphisms

PRIME RINGS

of linear

rank.

as semi-

are also

of rings,

characterized

of the rings

these rings

The first m e n t i o n e d

as an abstract

is also

characterizations

rings

of finite

to isomorphisms

the semi-

as well

socles

class

The nonzero and

found.

is to give

One of these to dense

in the second m e n t i o n e d

collect

section

rings

is

of the

transformations

space.

components

of their nonzero

semigroups

because

representation

characterizations

vector

atomic

of the latter

studied here

II.i The purpose

Semiprime

uniquely

rings

several

ways,

transformations

rings with minimal

and a m a t r i x

dimensional

extensions

are e x t e n d e d

Simple

into h o m o g e n e o u s

For a subclass

in several

as the ring of all linear

or a finite

is d e c o m p o s e d

socle

ways

additional

to certain

rings of linear

rank.

to give

as well

relative

are c h a r a c t e r i z e d

in several

is d e c o m p o s e d

title w i t h various

to dense

of finite

ring

group

socle

are used

transformations

socle of a semiprime

terized

a nonzero

IDEALS

and m a x i m a l i t y

they are isomorphic

These

linear

in the

atomicity

are c h a r a c t e r i z e d

for them.

ring of all

of rings

ONE-SIDED

If of

A = ~a},

A

and p r e l i m i n a r y theorem.

results,

We denote

by

we will ~

an

otherwise.

and

~

concepts

in a single

B

which

we write

are n o n e m p t y

subsets

are sums of elements aB

instead

of

of

let

AB

denote

of the form

ab

with

~a}B;

Ab

~,

has

a similar

66

meaning.

By induction, we define

that

is the set of all sums of elements of the form

An

si~

A n = (An-I)A

for

n >

i, w h e r e

A 1 = A, and see

sla2 .°. s

where

n

A11.1.2 DEFINITION.

if

In = 0

ideals.

for some

An (left, right,

n.

A ring

For s nonempty subset

~ B

two-sided)

is semiprime of

~,

b E B}

~r,~(B) = [r E ~ Ibr = 0

for all

b ~ BJ

Then

A

~] (~)

and

A semiprime ring L

is a two-sided ideal of

~

B

~/r(~)

~

be a left ideal of

N

is nilpotent

if it has no nonzero nilpotent

for all

It is clear that both

Let

of

the sets

are the left and the right annihilators of

II.l.3 LEMMA.

I

= [r E ~ I rb = 0

~,~(B)

PROOF.

ideal

in

3, respectively.

are nilpotent

ideals of

~.

has no one-sided nilpotent ideals. ~

such that

Ln = 0

and let

and a typical element of

An

A = I~.

is a sum of

elements of the form Pl

P2

Pn

(k~_-l~k_,irk,l ) (~k=l~k,2rk,2)

'''

(k~_-l~k,nrk,n)

'

But (~irl)(£2r2) ... (~nrn) = ~ i ( r i £ 2 ) . . . (rn_l%n)r n = 0 where

~i 6 L, r i E ~,

by hypothesis.

Thus

since

ri%i+ 1 ~ L

L ~ ~%(~)

where

and ~(~)

L n = 0. = 0

Hence

An = 0

by hypothesis,

so

A = 0

so that

L = 0.

The case of right ideals is snslogous. It follows from the lemma that the absence of nilpotent (2) right ideals,

(3) ideals are equivalent statements.

(i) left ideals,

We will be dealing mainly

with such rings.

if

11.1.4 DEFINITION. 2 A = A. Note that if

an

A

A (left, right,

two-sided)

ideal

is a minimal one-sided ideal of

ideal on the same side as

A; so by m i n i m a l i t y of

~, A

A

of

~

is idempotent

then

A2 ~ A

either

A 2= 0

and or

A2

is

A 2 = A.

Hence a minimal one-sided ideal is either nilpotent or idempotent° 11.1.5 LEMMA.

Every idempotent minimal one-sided ideal of

~

i_~s generated by

a__nnidempotent. PROOF.

Let 2

a E L, then La = L L.

L

since

Define

L

L

be an idempotent minimal left ideal of

= O, a contradiction, is a minimal

~ : x ~ xs

(x C L).

hence

La # 0

for some

3.

If a E L

La = 0

for all

and thus

left ideal. Consequently ea = a for some e in -I Then 0~ is a left ideal of ~ contained in L,

67

and is d i f f e r e n t Hence in

from

L

e 2 - e ~ 0~ -I

L, so

since

Le = L.

Hence

The following

~ e = L.

statement

11.1.6 COROLLARY.

e ~ O~ -I.

e 2 = e.

implies

will

By m i n i m a l i t y

Thus

Again

prove

In a semiprime

Le

the case

quite

of

L, we have

is a nonzero of right

0~ -I = O.

left ideal

ideals

contained

is analogous.

useful.

ring every m i n i m a l

one-sided

ideal

is gener-

ated by a n idempotent. II.l.7 NOTATION. denotes

by induction, with

If

A

and

the set of all elements we

a i ~ A. 11.1.8

nilpotent

that

An

left,

right or two-sided of

ab

subsets

with

of w r i t i n g

An .

S

ideal

remains

Further,

The a n n i h i l a t o r s

with

S

~z(S)

of a semigroup

a E A, b E B°

is the set of all elements

For a semigroup

definition

ideals.

are n o n e m p t y

the same convention

DEFINITION.

the semigroup nilpotent

see

Again

B

of the form

aB

of the form if

zero,

S, AB

Defining

An

ala 2 ... a n

A = ~a}

will

the d e f i n i t i o n

be used.

of a

the same as in the ring case w i t h

is a semiprime and

~r(S)

semigroup

are defined

if it has no

as in the case

of a ring.

Note group,

that in spite of the different

for a subset

A

11.1.9 LEMMA.

of

Exercise.

II.i.i0

DEFINITION.

ideal

L

of

S

different

of

S

from

L Let

Sa = L

if

S

~

semigroup

S

S

is semiprime.

with

such

in a ring or a semiAn = 0

in

nilpotent

a nonzero

left

(right,

two-sided)

are called

corresponding

subset

left ideal;

statement

twoideal

0-minimal).

Then a nonempty

is s m i n i m a l

~.

ideals.

left

(right,

ideals

or a ring.

0 # a E L The

An

zero,

no nonzero

theory

be a semigroup for every

of

if and only if

has no o n e - s i d e d

if it contains

(in semigroup

satisfying holds

definitions

in

In a semigroup

is m i n i m a l

II.i.ii LEMMA.

converse

An = 0

A semiprime

PROOF.

sided)

~,

is valid

L

the for right

ideals.

Let

PROOF.

The

L

the property

have

is a left any

ideal.

Let

L~

0 ~ a E L t, we have Conversely,

S, and

let

Sa = L

or

and

following

~(S)

suppose

0 ~ s E L. Sa = 0.

arguments

are valid

in the s t a t e m e n t be a left L = Sa ~ that

Then

S Sa

The latter

would be a nonzero

L~

for both

of the lemma.

ideal of so that

is semiprime,

S

that

L~ = L

and

let

nilpotent

cannot

ideal.

L

of

and a ring

It is easy

such

is a left ideal alternative

a semigroup

occur

contained since

then for

is minimal.

be s minimal S

to see that

0 # L ~ ~ L; L

S.

then

in

left L

ideal of and

a E ~(S),

thus

L

68

11.1.12

LEMMA.

A ring

[R

i_~s semiprime

if and only

if

~

is a semi~rime

semigroup. PROOF. an ideal I = 0

Necessity.

of

[R

snd

and hence

Let

~

be an ideal of

h~

with

by hypothesis

I n = 0.

implies

Then

C(I)

= 0.

But

ideals

of

C(I)

is then

is semiprime.

Sufficiency.

Obvious.

II.l.13 COROLLARY. and

I

[C(1)] n = 0, w h i c h

In a semiprime

ring

[R, minimal

one-sided

concerning

a n idempotent

[D0~ coincide.

II,i.14 LEMMA. semiprime i)

ring ~e

ii)

[R

following

left

is a d i v i s i o n

e~

is a m i n i m a l

PROOF.

i) =

group.

Let

e = ra

for some

statements

e~e

ii).

ring.

right

Since

ideal. ~e0{e = [Re ~ ~e~, we have

0 # s 6 ~e~e, then r E ~.

ere E e~e,

0 ~ a % ~e

ii) =

i).

(ae)ba # 0 y E e~e which

and since

that that

e

0 # ebs E

of ii) and iii)

II.l.15 Then

introduced

DEFINITION.

the subring

of

[R

[Ai}iEl,

and is d e n o t e d

be equal

to zero.

below Let

is an additive

that

[Ra = [Re.

Hence

[Ai}iE I

so

*ab~ ~ 0

e~e

is a d i v i s i o n

implies

which

of

~e

also

so that

ring,

there

exists

c = ce = cyeba E [Ra Ks c ~e

is a minimal

so that left

ideal.

by symmetry.

multiplicative, "division

hence

the same

ring" by ~'group with

to D i e u d o n n ~

statements zero".

The

[2].

be a n o n e m p t y @ A. iEl l

b E ~

implies

that

elements

and thus

for some

now follows

is due

the nonzero ring.

aba # 0

a E ~

entirely

by

e~e,

c E [Re, we obtain

I.ii

by r e p l a c i n g

generated

by

Since

Conversely, and

of

a = ae E a~

For any

0 # a E ~e

in a semigroup

notion of socle

implies

is a d i v i s i o n

Hence

e~e.

The proof of 1.14 is almost remain v a l i d

~e~e

Then

= e.

~ e c [Ra.

The e q u i v a l e n c e

so

is semiprime.

y(eba)

for every

e~e

= (ere)s

is the identity

group,

0 ~ a 6 ~e.

[R

and thus

such

proves

~a = ~ e

Let

since

and i.ii

that

Consequently

form a m u l t i p l i c a t i v e

([Ra) 2 # 0

of a

ideal.

e = ra = (er)(ea) where

e

are equivalent.

is a m i n i m a l

e~e

iii)

The

is called

family

of subrings

of

~.

a sum of the subrings

~ A i. A sum of an empty set of subrings is defined to iE I A sum of all m i n i m a l right ideals of [R is the socle ~ of [R.

69 Note

that

all the elements

r I + r 2 + ... + r n a right

where

ideal of

sum of all minimal

every minimal O # a E L.

that

by

inclusion = a(ex)

from

1.5,

i.ii,

where

~e~e

and

Let

of

~.

thus 0 } b E

proves

ideal

and hence

that

11.1.17

a 6 ~b~

from right

II.l.18

and

II.l.19

ideals

PROOF.

e

of

~

left

coincides

such

~

is

with

~

Thus

the

next

ideal and

that

that

so that exists

if

contains

~ e = L.

x E ~, we have

there

~

Further

is semiprime,

We prove

y E ~,

if and only

that

left

that

= 0, it follows

for some

ideal

to prove

since

a~.

for some

~(~)

proving

the opposite

b = ax = exD~ # 0

(ae)x and

thus

0 # exye E e~e, c E e~e

such

that

1.14,

~, we easily a form

= ae = a ~b~ = s~.

By i.ii,

a~

is a m i n i m a l

right

that

the set

theoretic

union

of all minimal

left

and

coincide.

left

ideals

socle.

is called

Because

interested

of 1.16,

in semiprime

rin$

~

is an ideal

the left socle,

whereas

we need not make

rings.

The

of

the

following

~. the socle

left-right

concept

is

[4].

DEFINITION.

A ring

LEMMA.

If

~

First note

~ = ~e~e &

act on

see

that

~ ~ 0 right

is a simple

spaces that

that every m i n i m a l

Letting

~

be a m i n i m a l

The socle of a semiprime

the risht

of dual v e c t o r

Define

that

therefore

is a sum of its minimal

~.

L

b~ c

: s(exyec)

the proof

w h e n we are

to J a c o b s o n

By

R i, and that

~ ~.

COROLLARY.

is called

e.

ideals

has a minimal it suffices

let

ring by 1.14.

The sum of all m i n i m a l

implies

~

Hence

It follows

a E ~

It follows of all minimal

(i.e., ~

right

the form

Consequently

which

(U,V)

that

'~R, then

b(yec)

distinction

are of

the socle

an idempotent 2 ~ ( *a~)a = (~a) = ~a

is a division

(exye)c = e.

~,

By symmetry,

and hence ex # 0. Since 2 } 0. Hence exye # 0

(ex~)

due

rin$

there exists

also holds.

also

above

1.14

ideal.

left ideal

aS # 0.

~

left ideals.

right

By

for some m i n i m a l

In a semiprime

It follows

it has m i n i m a l

~a = L

ri E Ri

socle

~.

II.i.16 LEMMA.

PROOF.

of a n o n z e r o

~2 ~ 0

right

implies

ideal

by

with

its socle

then

that

~

U = ~e

is a left and

U (U,&)

= e(rs)e

~ ~ Su(V)

fo__~rsome

pair

g.

is of the form

ring and

(er,se)

if it coincides

rin$,

rin$

on the left and on

(g,V)

~:V xU ~ &

atomic

over a d i v i s i o n

is a d i v i s i o n V

is atomic ideals).

is semiprime. V = e~

Hence

1.6

for some idempotent

is a minimal

left ideal of

on the right by m u l t i p l i c a t i o n is a right v e c t o r (r,s E ~).

Then

~

space over

in ~.

is e v i d e n t l y

70 bilinear.

If

and hence

erse = e # 0

(er,se) # O

er # 0, then

for some

(U,A,V;~) = (U~V) For every

e~

= ~

= V

r E ~

and hence

ers = e

(er,se) # 0. ~

for some

Dually

is nondegenerate.

se # 0

s E implies

Consequently

is a pair of dual vector spaces.

r E [~, let

r

be defined by

(es)r = e(sr) then it follows inm~ediately that

(s 6 ~); r ~ £(~,V).

~(se) = (rs)e then similarly

by 1.11, so

which shows that

r ~ £(U,g).

Define

r

by

(s E ~);

Since

((es)r, te) = (e(sr),te) = esrte = (es,r(te)) = (es,~(te)), it follows that

9

The m a p p i n g e ~ O, we have

)<:r ~ r

hence

(r E ~)

ker X # 0{

it follows that For any

is the adjoint of

Thus

r E b~, we have

rank ~ = i.

Since

in

U, and hence

X

I~

is simple and

ker X

is an isomorphism of

(er)e = ere = (ere)e

Su(V)

r E £u(V).

is easily seen to be a homomorphism.

and since

ker X = 0.

r

is an ideal of

where ~u(V),

Since

is an ideal of

~

into

~,

~u(V).

ere ~ &, e E V, and the set

T = {r E '~ I ~ E au(V)] must be a nonzero ideal of For

~.

Hence

[i,y,k] 6 $2~u(V), we have

T = ~

and thus

X

maps

u.l = te, y = epe, vl = er

~

into

Su(V).

for some

t,p,r E ~{, and thus (es)[i,y,~] = ( e s ) ( t e ) ( e p e ) ( e r ) = which shows that

(teper))¢ =

that

onto

X

maps

~

[i,~{,h].

~u(V).

e(steper) = (es)teper

Since

Therefore

~2,u(V)

generates

~u(V),

it follows

~ ~ Su(V).

The following concepts will play an important role. II.i.20 DEFINITION. a ring.

If

~

acts on

Let M

M

be an additively w r i t t e n abelian group and

on the right satisfying:

(x + y ) a

= xa + y a ,

x(a+b)

= xa+xb,

for any

be

x,y E M, a,b E ~,

(×a)b = x(ab), then

M

for w h i c h

is a .risht ~ - m o d u l e or simply an ~-module. xa ~ N

~ B ~_ ~, let the form

ab

AB with

for all

x E N, a E ~

is an ~ - s u b m o d u l e of

be the set of all elements of a ~ A, b E B.

An additive subgroup

M

M.

If

N

of

M

~ ~ A c_ M,

which are sums of elements of

71 An ~ - m o d u l e (2) M and

M

~

are denoted

r E 5,

there exists

11,1.21 module. AB ~

An c

ideals

a ring

A

B ~

I

or of

that

~

m E M M

and

since

II.i.23 semisroup ideal

of

K

and

Let

I

~

A

irreducible

and

is a prime

B

of

ring

if for any ideal

0

ring

is a semiprime

A

2,

if the of

~,

ideal

of

~.

ring and can be taken

if and only

and

B

M

as

if for any nonzero

mA = M

then

be nonzero

is faithful since

mA

0 = mAB = MB

Consequently

nonzero

ideals.

on

to

S

with

AB # 0 zero

ideals

and

then

K

S

that

K ~

of

~,

A ~ 0,

and

there

is an ~ - s u b m o d u l e which

and

of

is impossible

~

and a dense

ideals,

On

or K

since

is a prime extension

is c o n t a i n e d

ring.

of a

in every nonzero

such S

x = y.

define

It is easy

is the equality S

is a dense

A semigroup

the

I.

to see

relation.

extension

is completely

Thus

following

of

0 - ~

K P I = 0

relation:

since

x T y

if

that

T

is a congruence

Thus

T

must be

K, but

then

the

1 = 0.

if it is isomorphic

to

semigroup.

is not a customary

and Preston result

the Jscobson

Density

in G l u s k i n

but

definition

it turns out

of a completely to be equivalent

([i], Chapter 3, S e c t i o n

of this

are due to 3 a c o b s o n proved

A

is a s e m i g r o u p

x,y E I

theory of semigroups,

principal

if

Since

AB = 0,

be an ideal of

11.1.24 DEFINITION.

Clifford

If

S

restriction

relation

s Rees m a t r i x

This

~-module.

proper

nonzero

if either

whose

equality

element

tins is prime.

m A # 0, so that

If

without

Further,

of a semiprime

ring,

is faithful.

LEMMA.

has no proper

S

M

M

S.

PROOF.

and only

that

is irreducible.

B ~ 0

and

AB # O.

primitive

irreducible

such

I.

is a prime

~, we have

be a primitive

be a faithful

if it has a faithful

if for any ideals

if and only

Every

~

if for any nonzero

is .semiprime

the d e f i n i t i o n

II.i.22 LEMMA. Let

~

r E ~

the zero of both

mr # 0.

is primitive

A ~

is semiprime

of

that

m E M,

(caution:

is faithful such

ideal

I

for some 0

I.

with

note

B

and

is a prime

PROOF.

exists

on

A ~

~

Also

and

M

K

that

~

M

M

m E M

An ideal

is in conformity

a definition.

~

that either

implies

(i) mr @ 0

from

Further,

A ring

of

is prime.

Hence

0).

an element

I

if

different

DEFINITION.

implies 0

I

This

by

A n ideal

I

ideal

M

is irreducible

has no ~ - s u b m o d u l e

section.

Theorem, [4].

[4].

The

2).

0-simple

semigroup

to the usual

We are now ready

The i m p l i c a t i o n

ii) = v)

one,

to prove

is a special

[3].

implications

i) = 1.2.13 i) and vii) = 1 . 2 . 1 3 i )

implications

appeared

ii) ~ v) ~

in Petrich

the

case of

see J a c o b s o n

The remaining

The equivalences

in the see

[4].

1.2.13 i) are

72 11.1.25 THEOREM. i)

~

The followin$ conditions on a rin$

~

is a semiprime ring with a minimal right ideal

are equivalent. A

such that

~r,~(A) = 0. ii)

~

is a primitive rin$ with a nonzero socle.

iii)

~

is ~ prime ring with a nonzero socle.

iv)

~

is an essential extension of its simple socle.

v)

~

is isomorphic to a dense ring of linear transformations on a vector

space containing a nonzero t r a n s f o r m a t i o n of finite rank. vi) vii)

~O~

is

~

has a completely 0-simple

PROOF.

dense extension of a completely O~simple semigroup.

a

i) = ii).

We consider

ideal contained in every nonzero ideal of

A

as an ~-module.

Then its m i n i m a l i t y as a

right ideal implies that it is irreducible and the condition

~r,~(A) = 0

that it

is faithful. ii) = iii).

This is a special case of 1.22.

iii) = iv).

Let

E

be the socle of

be a minimal right ideal of

~.

If

I ~ ~r(~) = 0, a contradiction. is a prime ring.

Hence

exists an idempotent where

re E E

that

es E ~

Similarly

bra ~ 0

and

for some

ce # 0.

s E ~.

again by 1.17.

0 ~ A 2 = (c~)(~),

A = e~.

Further,

that

contained in

d E I

1

so that

that

dE c

idempotent, we have If

J

is an ideal of

is prime we must have iv) = v).

Letting

and thus ~

E

~

of

~u(V)

Su(V).

Thus

containing

Su(V).

since

By 1.6, there b(re)a # 0

c = bre, we see

(~c 'e~)2 ~ 0

that

so that

0 ~ d ~ A N I

since

the hypothesis yields u E ~.

It follows that

d ~ = A.

This shows that I = ~.

Since

dE

is

On the other hand, I

~

J ~ ~ = O, then also ~

contains every contains a nonzero

~ ~ ~u(V).

Finally,

J ~ = 0, and since

is an essential extension of

~.

~, by 1.19 there exists a pair By 1.7.13, £u(V)

By the ring v e r s i o n of 1.7.5, ~

of linear transformations.

Letting

A

is simple.

be the socle of

of dual v e c t o r spaces such that extension of

~

a ~ A. and thus

implies that

and thus

that

such that

J = O.

A

A ~ I.

~, which implies

~2 # 0

b(re) E I.

for some

~

implies

a = ea

~, and

and thus

(N I ~ ) ( ~ A ) # 0

b E I, r E ~,

i.ii together with

d(ud) # 0

ideal of

P~ ~ ~£(~) = 0

d = esc, we deduce

a nonzero right ideal of

minimal right ideal of

be a nonzero

Hence

The hypothesis

Letting

and hence

I

~ A # 0, and thus

for some

such that

by 1.17, which implies

0 ~ c E I

esc # 0

e

~,

~ I ~ = 0, then

(U,V)

is a maximal essential

is isomorphic to a subring

1.2.13 asserts that

~

is a dense ring

73 v) = vi). (U,V)

By 1.2.13, we have

of dual vector spaces.

dense extension of

~2,u(V).

0-simple semigroup

~2,u(V).

vi) = vii). calculation),

~ ~ ~,

where

Su(V) ~ ~J ~ ~u(V)

By 1.7.10, we have that Hence

~J

for some pair

gu(V) = ~£u(V)

is a maximal

is a dense extension of the completely

Since a Rees matrix semigroup has no nonzero proper ideals

the same holds for a completely 0-simple semigroup.

(simple

The desired

conclusion now follows from 1.23. vii) = i).

The hypothesis

question is unique.

Letting

in every nonzero ideal of implies that

~

see that for every

a right ideal of

since

~

A

diction.

~.

a E A

M

contains nonzero idempotents,

such that

since

er E R.

~I~.

If

B

~

is a right ideal of

~

Further,

J ~ O, then

~,

A

since

A

and thus

A

be a

B = A

A

is also

A, then or ~

B

is

B = 0. by 1.13

is a right ideal of ~, J = ~r,~(A) is 2 0 ~ A ~ AI C AJ = O, a contra-

~r,~(A) = 0. M

is defined in a dual manner

on the left.

to a (right) ~ - m o d u l e by making

The d e f i n i t i o n of a "left ~ - s u b m o d u l e " ,

"left primitive ring" should be clear.

left and right can be interchanged.

with

in

Hence for any

snd thus also of

I c j

right socles coincide in this case. ideal

a = ae.

contained in

"irreducible",

Now note that conditions vi)

and vii) contain no left or right modifier, which implies conditions

~

contained

the hypothesis Let

Consequently

which by m i n i m a l i t y implies that

is a minimal right ideal of

If

~

By a simple c a l c u l a t i o n in a Rees matrix semigroup, we e ~ ~

A left ~ - m o d u l e act on

~

is an ideal of

there exists

Therefore

"faithful",

Since

I

ar = (ae)r = a(er) E A

is semiprime.

an ideal of

~.

R.

also a right ideal of Consequently

that the completely 0-simple ideal

cannot have nilpotent ideals and is thus semiprime.

minimal right ideal of

r E ~, we have

implies

I = C(~), we see that

that in the remaining

Indeed 1.16 says that left and

In condition i) we may take a minimal

~%,~(A) = O; in ii) a left primitive

left

ring; in v) the linear trans-

formations can be taken on a left or right vector space. II.i.26 DEFINITION. the semigroup socle Exercise

g

The set theoretic union of minimal right ideals of of

~

is

~.

1.28,v) below implies that

after 1.16 that

~

~

is an ideal of

~IR.

We have remarked

agrees with the set theoretic union of sll minimal

left ideals

in a semiprime ring. II.i.27 COROLLARY.

A rin$

~

of dual vector spaces if and only if maximal relative to ii) or iii) semigroup socle.

i__ssisomorphic t___0_o£u(V) ~

for some pair

(U,V)

satisfies the conditions of 1.25 and is

or iv) for a fixed socle or vi) o__~rvii) for a fixed

PROOF.

This follows from 1.25,

1.2.13,

1.7.10 and 1.7.13.

II.i.28 EXERCISES. i)

Let

~

be a ring and

I

be a nonzero ideal of

~.

Prove the following

statements. (a) ~(I)

= 0

~(I) = 0

where (b)

~(I)

and

~

is an essential extension of

I

if and only if

= lr E ~ I rI = Ir = 0].

Mr(I ) = 0

and

~

is an essential extension of

I

if and only if

~r,~(I) = 0. ii)

Show that a ring

~

for which

~

is completely 0-simple must be a

division ring. iii)

Show that a prime ring with a nonzero socle and without nonzero nilpotent

elements must be a division ring (an element

r

is nilpotent

if

rn = 0

for some

positive integer n). iv)

Let

the ring

v) that

~

and

~

in 1.25.

denote

the semigroup socle and the socle, respectively,

~ = C(~)

(b)

The sets of minimal

(c)

Every

Let

A

Let

0 # a E A.

A

and is contained in every nonzero ideal of

be a minimal 0

~

right ideal and

~. ~

coincide.

is also a (left) right ideal of a

be an element of a ring

or a minimal right ideal of

13],[4],15].

~.

Show

~. ~

and let

for this section are Dieudonn~

[2],[3], G l u s k i n

Further material concerning primitive rings and rings

of linear transformations can be found in Behrens

([i], Chapter II, Section

i),

(17], Chapter II, Section 2 and Chapter IV, Sections 9, 15), Kaplsnsky

Part II, Section 6), McCoy Section 2), Wolfson

~.

aA = a~ = A.

The main original references

Jacobson

R, ~, and

be an idempotent minimal right ideal of a ring

Show that

[4} and J a c o b s o n

(left) right ideals of

(left) right ideal of

is either

of

Prove the following statements.

(a)

aA vi)

~

([I], Sections

20,21,23-27), R i b e n b o i m

([4],

([i], Chapter III,

[i],[3],[4].

II.2 After some preliminaries, sided ideals in several ways.

SIMPLE RINGS

we will characterize simple rings with minimal oneWe will supplement these results by establishing

further properties of dual pairs of vector spaces in connection with these rings.

75

II.2.1 product a k x

DEFINITION.

~ o p and

If

x p b

for some

II.2.2 DEFINITION. i)

a~b

ii)

Then

~

£, R, ~

are b i n a r y on

X

relations

given by:

on a set

a k o p b

X,

their

if and only

if

x E X. S, define

the following

relations.

= b U Sb.

a U aS = b U bs.

are called Green's

Clearly

£

left

generate

then

p

~ = ~ o ~.

principal b

and

relation

On a semigroup

a USa

aR b ~

iii)

k

is the b i n a r y

L(a)

and

~

are equivalence

ideal of

S

generated

the same principal = Sa.

relations.

left

If

a

statements

In a Rees m a t r i x

The set

a, so that

ideal.

The c o r r e s p o n d i n g

II.2.3 LEMMA.

relations.

by

hold

semisroup,

a~b

L(a)

= s USa

if and only

has a left identity, for right

is the

if

a

say

and a = ea,

ideals.

any two nonzero

elements

are

~-related. PROOF.

For

the v e r i f i c a t i o n

S = ~ that

o

(I,G,M;P)

(i,a,p) £

and

(i,a,p),(j,b,~)

(j,a,D)

and

E S

where

(j,a,~) ~% (j,b,v)

a # 0, b # 0, is left as an

exercise. II.2.4 a function of abelian

DEFINITION. ~

groups

a E A, r E ~. and

B

(respectively

A

and

If

are said

II.2.5

If

mapping

~

A

and

into

B

B

are ~ - m o d u l e s

(respectively

is an ~ - h o m o m o r p h i s m

(~a)r = ~(ar) is one-to-one

(respectively and onto

to be ~ - i s o m o r p h i c ,

r(a~)

it is called

to be denoted

if

~

left ~ - m o d u l e s ) ,

is a h o m o m o r p h i s m

= (ra)~)

for all

an N - i s o m o r p h i s m

by "A ~ B

and

A

as ~ - m o d u l e s "

as left ~ - m o d u l e s ) .

LEMMA.

The

followin$

conditions

on idempotents

e

and

f

of a ring

N

are equivalent. i) ii)

e~ ~

f~

N e --~ N f

iii)

There

iv)

e g f.

PROOF. Then

as left N - m o d u l e s .

exist

ii) =

a E ~f,

as N - m o d u l e s .

a,b E ~

iii).

b E Ne

such

Let

~:Ne ~ ~f

f =

since

ab = e, ba = f.

be an N - i s o m o r p h i s m ,

and

-i -i e = eqxp = a%0 =

-i %0

that

f~-~

=

~

=

(af)~ -I = a(f~ -I) = ab,

(be)~

is also an N - i s o m o r p h i s m .

= b(~)

=

ba,

and

a = e~, b = f q - i

76 iii) =

iv).

Letting

c = fbe, we obtain

e = ab = a(ba)b = a(fbe) which

implies iv) =

and also

that

ii).

e £

Let

c.

One

shows

e £ b, b ~

b = fb = be.

f.

Letting

and

= ae E ~ c

similarly Then

that

e = xb

c ~

and

f

so that

f = by

e ~ f.

for some

x,y E

a = xf, we obtain

ab = (xf)b = x(fb) ba = b(xf)

c ~ ~e

= xb = e,

= bx(by)

= b(xb)y

= bey = by = f,

a = af. Then of

the m a p p i n g s ~e

r~

into

~ : r ~ ra

~f

and

= rab = re = r and

~

(r E ~e), into

~e,

and similarly

~:r ~ rb

(r E ~f),

respectively.

r@~ = r

For any

for any

are ~ - h o m o m o r p h i s m s r E ~e

r ~ ~f,

which

we have proves

that both

are ~ - i s o m o r p h i s m s .

The equivalence II.2.6 function

~f

of i) and iii)

DEFINITION.

P =

(phi)

row-independent

now

follows

For a d i v i s i o n

mapping

A x I

if for any finite

ring

into set

by symmetry.

&

g

and nonempty

sets

I

is a A × I-matrix

over

g; F

kl,k2,...,k n E A

and

A, any is left

and scalars

n

O 1 , O 2 , . . . , O n ~ g, o I = 02 =

... = o n = 0; right

II.2.7 division

the relation

DEFINITION

ring,

p =

(cf.

(p%i) &; by

over

having

a finite number

(aik) + (bik) =

(aik + b i k ) on the right

is possible

exercise

to show

over

with

£

Caution. We are

ready

precisely

and iv)

in the next

by a m p l i f i e d version

of the Rees

Section

2).

a ring

for

the m a i n

the rings

on

gu(V)

closely

theorem

The r e m a i n i n g

here

to this,

by Hotzel

in semigroups,

the

of m a t r i c e s

it the Rees m a t r i x

ring

rings".

pair [4].

is due

It follows

easily

ideal

up to an

(U,V).

are,

This was

The e q u i v a l e n c e to J a c o b s o n

[l] and its proof

see C l i f f o r d

are new.

£

ring.

one-sided

time;

over

by entries

It is left as an

of this section.

for the first

equivalences

addition

B).

by G l u s k i n

be a

A * B = APB, where

"Rees

for some dual

&

I ×A-matrices

we call

a minimal

related

established

appears

and

is called

[2] and r e d i s c o v e r e d

theorem,

A

sets,

column-independent

multiplication

a Rees m a t r i x

result

analogously.

with

given by

in this way:

rings w i t h

implies

be nonempty

together *

I

and right

row-by-column

class of rings

of i) and v) was somewhat)

A

the set of all

entries,

p, or simply

that simple

by D i e u d o n n ~

denote

i 6

is defined

and

of the hypothesis

matrix

established

I

is the usual

An u n r e l a t e d

1.2.13

Let

of nonzero

that we obtain

sandwich

isomorphism,

equivalence

1.4.4).

and m u l t i p l i c a t i o n

because

finally

from 1.25 and

column-independence

~(I,g,A;P)

multiplication (which

for every

be a left r o w - i n d e p e n d e n t

A × I-matrix only

~ o.p. i = 0 j=l ] ~.j

The

(due to Hotzel

it represents and preston

[3].

of i)

the ring

([i], C h a p t e r 3,

77 II.2.8 THEOREM.

i)

The followin$ conditions on a rin$

i__s_s~ simple ring w i t h !

ii)

is a prime

iii)

(respectively ~rimitive)

is isomorphic

v)

~

vi)

~

i) = v).

Since

v) = iv). A = e~

satisfies

~

and

First let

right ideal of

But then

A

since

A = e~

let

I

follows that

C(~)

~

e E ~.

If

A = e~

A

~,

and thus

I~ = ~ I = O.

contains its negative since

right ideals.

A

is a

and thus the equality ~, minimal

I [~ R = 0.

It

We have just seen that every

is a minimal right ideal of ~

~.

Hence with every element

is the union of its minimal

It follows that the elements of ~ are sums of elements in R. n t = ~ t~ with ti ~ ~ and n is minimal. i=l

Let

have the property that

hypothesis exists

then

w h i c h implies that

[E)l and suppose that

a

0 ~ t ~ I

and thus

We know by 1.3.11

r ~ ~,

is a right ideal of

~

~

D~

R.

be a nonzero ideal of

the ideal

~.

and

~ e(~L~) c A

which implies that

I ~ = RI = 0

a C A

minimal right ideal of ~,

has a

is a nonzero ideal of

is an ideal of

~0~. C o n s e q u e n t l y

Hence

condition iv) of 1.25, [E~

be a minimal right ideal of

in view of its m i n i m a l i t y in

of

~ = C(~).

~.

for some idempotent

ar = (ea)r = e(ar) E A

prevails.

~

~.

by simplicity of

Next

for principal risht ideals.

is isomorphic to a Rees matrix ring.

completely 0-simple ideal = C(~)

atomic ring.

to a dense ring of linear transformations of finite rank.

has a completely 0-simj~le ideal

PROOF.

that

minimal__ right ideal.

is a simple ring satisfying d.c.c,

iv)

are equivalent.

I ~ R = 0

e.~ ~ ~

implies

such that

that

n >

t i = tie i

i.

for

Since

~

I < i < n.

0 = (tl+t2+...+tn)[el+e2+...+en-

is completely 0-simple,

The there

Thus

el(e2+ ...+en) ]

n

n

= t + i=2 ~ t.(e~ • i + " . ' + e i -i + e i + l + ' "" + e n ) - i~=2tiel (e 2 + " ' ' + en)' so that n t = i~=2[i[el(e2 + ... + e n ) - (e I + ... + e i _ 1 + ei+ 1 + ... + e n ) ] where all the summands are in I N ~ # 0, and since ~

I.

I n ~

Consequently

R

and 1.2.13 implies that (U,V)

of vector spaces.

~,

c o n t r a d i c t i n g the m i n i m a l i t y of

is an ideal of

But

where

Hence

~

Su(V) ~ ~' ~ £u(V)

Su(V) = C(~2,u(V)) ~,

n.

Therefore

I n ~ = ~

is contained in every nonzero ideal of ~ ~ ~i

unique completely 0-simple ideal of ~' = ~u(V).

~, we must have

and thus

[E~, which by 1.25

for some dual pair

by 1.2.6 and

~2,u(V)

which by hypothesis implies

is the

that

is a dense ring of linear transformations of finite rank.

78 iv) = iii). Su(V)

Agsin by 1.2.13,

satisfies

d.c.c,

iii) = ii). ideal

A.

If

The hypothesis

B

0 ~ (b) ~ B ~ A

primitive)

implies

each implies

(b)

that

By 1.2.13,

we hsve

ring

of

zxs = onto

~(V)

~ o~ z .

the ring

~

g.

z

and

~

let

in

U.

o~

For sny

oX~ # 0

W = [wi}i~ I

right

hss the property

As above,

we conclude

thst

(respectively

in an atomic ring,

~

is also simple.

X E A

of dusl vector

Z = ~z }%EA

of

~:a ~ ( ~ )

A × A-matrices

s E £(V),

(U,V)

by letting that

~

V

snd represent (aE~(V))

is an isomorphism

over

&

the subspace is equsl

and

under the usual

of

to

of

V

spanned by the

Va, so that

identi-

be a basis of

of

U, let

A

have nonzero

a ~ ~u(V)

and

elements}. b

(i)

be the adjoint

We set

be as above

where and

P~i = (z~'wi) bw. = i

(i E I, ~ E A).

~ w ~.. it follows j jl'

that

(z a,wi) = (zk,bwi)

jEI (~O%

z ,wi) = (z~,~wjTij)j

~ox~P~i= ~ i P ~ j T j i

tation, we obtain that for the linear ~u(V) = [ A E ~ I

AF = PB.

"

We let

there exists

A = (o%),

Conversely,

transformation

s

(i E I, ~ E A)

that

~u(V) = £u(V)

B = (~ji) , so in the matrix no-

the last formula,

we must have

a column finite

going backwards,

a E ~u(V).

I × 1-matrix

such that

~u(V).

b

hence

in 1.6.12

for some

implies

Recslling

by

for a pair

&

a finite number of columns

P = (p~i)

so that

principsl

we have

~(V) = ~ A E ~ l o n l y

Letting

a minimal

are equivalent

over

of all row-finite

for which

~(V)

Now

A.

We fix a basis

We have remarked

addition and multiplication.

a

1.3.15 ssys that

0 ~ B c A, then for any

is also a prime

~ ~ ~u(V)

ss A × A - m a t r i c e s

of

of

~

that the socle of the ring is simple,

the elements

fying

contains

generated

the minimality

1.25 implies

spaces over a division

vectors

~

such that

By 1.25, prime and primitive

iv) = vi).

£(V)

Further,

ring~

ii) = i).

where

that

~

right ideal

contradicting Further,

~ ~ ~u(V).

right ideals.

is a right ideal of

0 # b ~ B, the principal

is atomic.

we have

for principal

B

AP = PB}.

implies

Consequently

over (2)

~ ~(V), we have by (i) and (2) the form of matrices

in

79

n

Suppose all

i E I

that

n

j=l~~'pX j nji = 0

for every

i E I.

Then

j~==l(~jzXj,wi)

= 0

for

( ~ o.z.,u) = 0 for all u E U. By the nondegeneraey of the j=l J J n form, we must have ~ o.zx = 0, which by linear independence of the j=l 3 j

bilinear

and hence

z%. implies Ol = o2 = ''" = On = 0. Thus p is left row-independent; J one verifies analogously that P is also right column-independent.

vectors

For u

a E ~u(V),

= ~w.T . ik j I Jl k'

v

z~a =

by 1.2.6, we have

a = k=l[ik,~k,Xk]~ .

Letting

= ~Jo, z , we obtain k D Ak~ D

k=l (zx' Uik)~kVXk

=

~ (z~,~w.T. i )~k(~OX z ) = ~ ( p ~ j T J i k ~ k O X k ) z D. k=l J J J k D k ~ k,j,B

(3)

As above we let A = ( o X ) where z a = ~ z , and now form a matrix n M A = ( ~ T.i ~k°X )' Then M is a I x A - m a t r i x over & having only a finite k=l J k k~ A number of nonzero entries; in matrix notation (3) becomes PM A = A. Suppose that C = (ci~)

is an I X A - m a t r i x

for which

PC = A

and let

over

A

with only a finite number of nonzero

M A = (mi~).

Then

P(M A - C) = 0

which

entries

implies

that

• Phi (miB- ciD ) = 0 for all ~,D E A. Since at most a finite number of mid ci~ ~i may be nonzero, this is a finite sum, and thus the right column-independence of P implies

that

equation

mi~ = ciB

PM A = A

for all

uniquely

Define a function

i,D.

Consequently

determines

~:~u(V) ~

C = MA

which

shows that the

M A.

[~(I,A,A;P)

by

~:A ~ M A.

For

A,A ~ E ~U(V),

we then have AA ~ = (PMA)(PMA~)

= P(MAPMAI ) = P(M A * M A ~ ) ,

AA ~ = PMAA ~ , which

implies

that

ring homomorphism. finite,

MAAI = M A * M A ~ . Let

M E

~(I,~,A;P),

only a finite number of columns

AP = (PM)P = PB

where

B = MP

from (i) and (2) above that onto.

Similarly

If

M A = 0, then

is an isomorphism

of

then for A

A @ ~u(V)

onto

= M A+MAt

and also

entries,

IX I-matrix over

4.

M A = M, which proves

and the kernel of ~(I,~,A;P).

so that

A = PM, we see that

contain nonzero

is a column-finite

A = PM A = 0 ~u(V)

of

MA+At

~

is trivial.

~ A

is a is row-

and It follows that

~

is

Therefore

80 vi) = i). of

~

and

Let

A

N =

~(I,A,A;P)

be a nonzero

way that the nonzero

rows of

labeled

We fix

1,2,...,n.

be a Rees matrix

element of A

J.

are labeled

k E I

and

ring.

We may choose 1,2,...,m

~ E A

Let

J

m n i=l~P~i (X=I~aiXP'Ak ) = 0

i < i < m 1 < X < n buk ~ 0

since

P

since

P

c

b lk =

is left row-independent,

C*A*D

i E I, % E A, we denote by entries

above and letting

only in the (i,h)-position,

form

(g)ih'

A

'~

the latter

minimal

(a C A)

whose

(i,h)-entry

the notation

~ c p a p d p,t,6,s jp pt t@ 6s so

to

that

J

contains

is nonzero Now

all matrices

c of the

~.

imply that there exists

that

(pki)i h

is a nonzero

With the same notation,

Phi # 0.

Then

~ If

~

~

that

For

is a completely

g

Conse-

A

onto

that

-i (pxi)i

-l (pxi)i k. *~

is a

is semiprime. O-simple

is a simple atomic ring and an ideal of a cardinal

the ring of all ~ X ~ - m a t r i c e s

of

-1 Aik = (pxi)i k * ~ *

ring which by 1.14 implies

since

idempotent.

the mapping

is easily seen to be an isomorphism

is a division

II.2.10 NOTATION. denote

with

c # 0, D = (1)kl, we obtain

It is also easy to verify

right ideal of

C(~)

P

showing

ring.

II.2.9 COROLLARY. then

Consequently

eix = c t,6 ~ PDta @p@k = CbDk.

which shows

must be equal

= (Pxi ih

AiX = { (a)ik I a E A} . Hence

i <_ i _< m,

-1

is a simple

-i )<:a ~ (Phia)i~

If

for

for

A # 0.

the l x A - m a t r i x

ej(~

-i)

(Phi)ikP(Phi)ik

ai% = 0

Continuing

where

since

on the matrix

-i

quently

J

(c)ix

in which case

arbitrarily

But then

The conditions

-1

but then

are

x~=nlPBiaikPhk •

n h=l~a.lhpkk = 0

that

contradicting

are zero.

C = (c)i D

= CPAPD = (ejo) = (cb k)i h

can be" chosen in

columns

k E I, ~ E A.

and all the remaining

introduced

implies

is right column-independent;

for every

For any is

which

in such a

and the nonzero

and let

i=l b k = 0, then

be an ideal

the notation

over

number and A

ideal of

~

for a rin$

~,

~. A

a division

ring,

let

A

>t

with only a finite number of nonzero

entries. This is an extension over

A.

Further,

A~

identity gxx-matrix. identify

U

with

V*

~

where

An

for the ring of all n × n-matrices I

has cardinality

We have seen in 1.3.12-3.15 and have

know for which dual pairs For convenience,

of the notation ~(I,A,I;Q)

that for

~(V) = ~u(V) ~ Adi m V .

(U,A,V)

we introduce

is

SU(A,V)

and

Q

is the

dim V < ~, we can

It is then of interest

isomorphic

a new concept.

~

to

to a ring of the form

81 11.2.11 DEFINITION. subsets if

lu i l i E l j

(vi,uj) = 6i j

Let

and for all

11.2.12 NOTATION. ~, respectively

form

G of n i =~Irigi

M

If

senerates

M

The usual definition

i, and let

ii)

~ ~ r ~

iii)

and

V

of a completely

M

can be written

in the

r.z E ~, gi E G. with zero, an idempotent

is

to the ordering

0-simple

(U,~,V)

semigroup

idempotent,

S

is that

cf. 1.24.

S

has a

The next

12].

be a pair of dual vector

then the following

statements

and a cardinal

have a common set

G

of ~enerators

have biorthogonal

primitive

generates

then

& ~ F

spaces with

are equivalent.

number

3.

consistin$

o f primitive

~.

ii) = iii).

A.

biorthogonal

A symmetric Suppose

that

dim U = dim V = 3. ~ = F

3.

E = (E i I i E I}

consists

A ~ F shows

that

that

generates

and

The hypothesis G.

with

For every of

be the non-

n ~ AE. which proves k=l ~k generates ~.

A = E

both

~

and

We will

~,

that

and let

show that

Z

and

W

are

and let

x

be

U, respectively.

be a nonzero vector

[ik,~k,X k ] E

• " "n Zl,Z2,...,z

and let

W = [u i I [i,~,X] ~ G}. V

&

has cardinality

proof shows G

over

I

Let

where

are I X 1-matrices

that the set

A simple argument

(x,ui) = o.

a k E ~,

of

entries,

It is clear

idempotents.

bases of

v = ov

bases. and

The elements

Z = [vx I [i,~,k] ~ G],

some

relative

of

then a

e = ef = fe.

Let

E i = (1)i i.

zero columns of

Let

m

right) ~-module,

~

i) = ii).

i E I, let

such that

right, multiplication).

(respectively

if every element n i =~Igiri) for some

only a finite number of nonzero

E

the

the left and the right ~-module

ring

i f i) occurs,

PROOF.

orthogonal

Then

are biorthogonal

idempotents.

U

Moreover,

denote

is a left

~ = ~U(g,V),

~

V, respectively,

is left, respectively

for some division

and

orthogonal

and

[i], see also Dieudonn@

11.2.15 PROPOSITION.

i)

U

proper ideals and a primitive

is due to Hotzel

dim V >

~

and minimal

e < f ~

result

of

In a ring or a semigroup

if it is nonzero

zero, no nonzero

and

M

(respectively

11.2.14 DEFINITION. primitive

~

the action

11.2.13 DEFINITION. subset

be a dual pair of vector spaces.

i,j E I.

Let

(i.e.,

(U,V)

[v i l i E I]

Hence

in

V.

implies

Choose any that

ui E ~

[i,l,~] =

~ ak[ik,~k,k k] k=l

for

82

v = uvD = (x,ui)v D = x[i,l,~] n

n

= X(k~__laklik,~k,Xk]) proving that Let

Z

generates

= k~=i(xak,Uik)~kVhk ,

V.

[i,v,X],[j,6,D] E G

be distinct.

[i,~,h] [j,~,D] = so that

(vX,uj) = (vD,ui) = O.

similarly

X # ~.

and w r i t i n g

all

VXk.

[j,~,D] [i,~,X] = 0

Since

(v%,ui) # O, it follows that

that

vx =

There exists

i

and

W = ~w k I k ~ K}, we also have

n ~ ~,v , where k=l ~ h k such that

v

v E Z h' X k

and

that

v~

and

i # j

This establishes a one-to-one c o r r e s p o n d e n c e b e t w e e n

Z = ~Zk I k E K}

Now s u p p o s e

Then

is

Z

and

W,

(wi,zj) = 6i j.

different

from

[i,(vX,ui)-l,x] E G; thus

n 0 ~ (v~,ui) = k~=lOk(Vhk,Ui) = 0 by the above,

a contradiction.

This shows that

proJing that

Z

V.

Hence

Z

and

iii) = i).

is a basis of W

Z

is linearly independent,

A similar proof shows that

W

thus

is a basis of

U.

are b i o r t h o g o n a l bases. In the proof of "iv) = vi)" in 2.8, we take the given biorthogonal

bases and immediately obtain that

P = (phi) = ((z~,wi))

and the rest of the proof shows that

is the identity matrix

~U (g'V) ~ gdim V'

We finally prove the last statement of the proposition. Hence assume that -i Let e = (pxi)i h where Phi ~ 0. We have seen in the proof of

~U(g,V) ~ F .

"vi) = i)" in 2.8 that F F

&~ = e * F

is a simple atomic ring. ~ ~U l(A ,V ~) , w h e r e

Thus

U~ = F

SU(~,V) ~ ~ U ~ ( g ' , V I)

so that

& ~ F.

* e

is a d i v i s i o n ring isomorphic

Now 1.14, *e

and

to

r

and that

together w i t h the proof of 1.19, shows that V I = e*F

w i t h the induced bilinear form.

w h i c h by 1.5.15 implies that

We have seen above that both

biorthogonal bases w h i c h evidently implies that

(U,V) dimU

and

g ~ &t (U~,V j)

= dimV

and

and

dimV

= dimV I

must have d i m U ' = d i m V '.

Hence all four vector spaces have the same dimension. It is easy to see that

consists of all matrices in F all of whose nonth zero entries are situated in the i row. Let I be an index set of rows and columns of matrices entries

in

in the columns

F , and let

cjP~i = 0

for V~.

then

m ~i~=l(cjPli)i~j

i < j < m; but then It is clear that

dim V = dim V ~ = ~.

B = [(1)i D I ~ E I}.

DI,N2,...,~n,

m i~=l(Cj)i~ * (1)iBj = O, then

basis of

Vt

B

If

A E V~

has nonzero

n 1 A = k=l~(aiDkP~i)iXA * (1)i~ k" = 0

If

which evidently implies that

c. = 0 for 1 < j < m. Consequently B is a J is of cardinality ~, which finally yields

83 11.2.16 COROLLARY. and only if

~

A rin$

~

i~s isomorphic to a rin$ of the form

is a simple atomic ring for which

~

and

~

&

if

have a common set of

generators consisting of orthogonal primitive idempotents. 11.2.17 COROLLARY.

If

Note that in the case and

V

FQ ~ &b' then dim V < ~

that

U

that

~u(V) = ~u(V) = £(V) ~ gn

and

F ~ &

and

Q = b.

(U,V)

is a dual pair, 1.2.3 implies

admit biorthogonal bases, confirming what we already know, viz., for

n = dim V

snd

g

a division ring.

can be carried one step further as the next proposition, due to Msckey II.2.18 PROPOSITION. dim U = di_~mV PROOF. V and

and

Let

is denumerable, Let

(U,V) then

be a dual pair of vector spaces such that U

and

Z = ~zi I i = 1,2,3 .... }

U, respectively.

Jo = min~j I (zj'w I) # Oj

V and

admit biortho$onal bases. W = ~wi I i = 1,2 .... }

We will construct biorthogonal bases

Y = [Yi I i = 1,2,...]

of

V

and

and set

This

[2], shows.

U, respectively, by induction.

Xl = (Zjo'Wl)-iZjo '

be bases of

X = ~x i li = 1,2,...} Let

Y l = Wl' hence

(xl,Yl) = i. Assume that (xi,Yj) = 6ij

Xl,...,Xk, and

for

Jk = min ij I zj and set

Xk+ I = Zjk

i < i,j ~ k.

yl,...,y k

have been

Suppose first that

chosen in such a way that k

is odd.

Let

is not contained in the subspace generated by x I ..... Xn} k i~=l(Zjk,Yi)X i.

Then for

1 < t < k, we obtain

k (Xk+l,Yt) = (Zjk,Yt) - i=1~(Z.jk,Yi)(xi,Yt) = (Zjk,Yt) - (Zjk,Yt) = O. Next let

Jk+l = min~j I (Xk+l,Wj) ~ 0}

and set

k Yk+l = (W.jk+l - i~=lYi(Xi'Wjk+l))(xK~i .... W.jk+l)-l. For

i ~ t < k, we have k

-i

(xt'Yk+l) = [(xt'wjk+l ) - i~='l(xt,Yi)(xi ,w-Jk+l )](Xk+l'Wjk+l ) -i = [(xt'wjk+l) - (xt'Yt)(xt'w'jk+l)](Xk+l'Wjk+l)

= 0

and also k (Xk+l'Yk+l) = [(Xk+l'Wjk+l ) - i~__l(Xk+l,Yi) (x.l ,w.Jk+l )] (Xk+l,Wjk+l)-i

=

i.

84 If Yk+l

k

is even, we interchange

This process defines

that the sets X

X

generates

of

V

V

and

Y

the roles of

the sequences

spanned by some

xi's. V

Similarly

and

Y

II.2.19 PROPOSITION. (U,V)

that

SU(h,V)

Let

We construct a sequence

K =

is a subring of Let

I

en~en c_ I But then

Since

That

Hence

X

and

Y

are

to

h.

K.

[i].

~U(h,V)

for a dual

to

h

where

~ < ~

.

be a set consisting of nonzero elements of of idempotents of eI

~

such that en+ l

as follows. b I E e~el;

be an idempotent

By the suppose

for which

K

for

it follows

that

K

is the desired subring.

Then

I ~ e~e n # 0

n I N ek~e k @ 0

and hence

n = 1,2,...,

Next

and thus n by 1.3.18 and the last ring is simple by 1.3.6.

en3en ~ hrank e I

We

h, then every denumerable subset

isomorphic

e ~e c en+l~en+ n n 1

We will show that

argument implies that idempotents,

~

el,e2,..,

be a nonzero ideal of

el3e I ~

and

this is possible again in view of the proof of 1.3.20.

3.

since

U.

This is due to Hotzel

have been defined and let

U e ~e . n= 1 n n

Xk+ 1

One shows

is contained in the subspace

be any ring isomorphic

there exists an idempotent

bn+l,e n E e n + ~ e n + I, let

~

B = [b i li = 1,2,...]

el,e2,...,e n

to define yl,Y2, ....

is a locally matrix ring over

of vector spaces over a division ring

proof of 1.3.20, that

Let

is contained in a subring of PROOF.

3.

zk

generates

are now in a position to extend this result.

3

V

and

U, respectively.

Recall that 1.3.20 asserts

pair

and

are linearly independent as in the proof of 1.2.3.

follows from the fact that every

b i o r t h o g o n a l bases of

of

U

xl,x2,..,

ek3e k ~ I, so that

for

I = K.

for some

e

k = 1,2,..., which by the same Since

K

contains nonzero

it must be simple.

Next let

A

be a minimal right ideal of

an idempotent

e

so that

el~e I.

A = e(el~el) = e~e I.

division ring, and it is clear that

By [.6,

A

Hence by 1.14,

e(el~el)e = e~e ~ A.

is generated by e(el~el)e

is a

Furthermore,

e~e = e(el~el)e c_ eKe c_ e~e so that

eKe = e~e ~ h.

K, proving that is isomorphic to over

eKe

K

SKe(eKe,eK)

where

eK

is a minimal right ideal of

The proof of 1.19 now shows that

(Ke,eK)

K

is a dual pair of vector spaces

with the b i l i n e a r form induced by ring multiplication.

Note that

eK = e(n=iO en3en ) = nU=l(e~en ) ' =

left vector space over en3en ~ hrank en of

But then 1.14 implies that

is a simple atomic ring.

e~e

for

where

n = 1,2, ....

according to 1.3.18.

e~el, we may extend it to a basis

e~en c_ e~en+ 1

and

e~en

is a

Further,

dim e~e = rank e since n n S t a r t i n g with a basis V l , V 2 , . . . , V r a n k el

Vl'V2''''' v rank el'''''Vrank e 2

and with an obvious inductive process construct a denumerable basis of

of eK.

e~e2, The

85 same c o n s t r u c t i o n applies

to the right vector space

= dim Ke = dim eK ~ ~o" and

eK

If

~

have biorthogonal bases,

from 2.18.

In either case,

Ke, which shows that

is a finite cardinal, and if

~

then 1.2.3 says that

is infinite

2.15 implies that

Ke

the same conclusion follows

SKe(eKe,eK) ~ &

with

~ ~ ~o"

II.2.21 EXERCISES. i)

Show that in any semigroup Green's relations

~

and

~

commute and that

is an equivalence relation. ii)

Let

~ = ~(I,g,A;P)

be a Rees matrix ring.

a)

Find a Rees m a t r i x r e p r e s e n t a t i o n of the semigroup socle of

b)

Characterize

the right ideals of

c)

Characterize

the elements in the semigroup socle of

d)

Simplify the results in a), b), c) in the case that

matrix to obtain as simple characterizations iii) if

Show that in a regular ring

~e~e

is a division ring.

iv)

Let

£(V)/~(V) ideals.

V

~,

~.

~. ~. p

is the identity

as possible.

an idempotent

e

is primitive if and only

be a vector space of countably infinite dimension.

Show that

is s simple regular ring with identity and without minimal o n e - s i d e d Give

an example of a simple ring without an identity and without minimal

one sided ideals. For further results on this subject, Dieudonn~

[2], Hotzel

Rosenberg

[i], W o l f s o n

[i], Jacobson

see Behrens

([i], Chapter II, Section 5),

[3],([7], Chapter IV, Section 15), Pless

[i],

[1],[3],

II.3

M A X I M A L PRIME RINGS

The theorem of this section gives a number of abstract characterizations of the ring of all linear transformations of finite rank on a vector space.

The remaining

results deal with further special cases of primitive rings with a nonzero socle. It will be convenient to introduce several new concepts. 11.3.1 DEFINITION.

A semigroup

isomorphic to an ideal of ~

for some ring

N~).

II.3.2 LEMMA.

If

simple atomic rin$ PROOF.

Let

K

K ~

~

S

is an r i - s e m i s r o u p

for some ring

~

so that

be an ideal of

K

(ring-ideal)

(equivalently,

S

is a completely 0-simple ri-semigroup, such that

K

is the semigroup socle of ~b~ ~.

Then

by 2.8, and in view of 1.25 it is clear that ideal of

~

K

~ = C(K)

if

S

is

is an ideal of

then there exists ~

and

~ = C(K).

is a simple atomic ring

is the unique completely 0-simple

must be the semigroup socle of

2.

86 II.3.3 DEFINITION. to a proper

A simple atomic ring is r-maximal

if it is not isomorphic

right ideal of a simple ring.

II.3.4 DEFINITION. not isomorphic

A completely

0-simple

ri-semigroup

to a proper right ideal of a completely

is r-maximal

0-simple

We are now ready for the first main result of this section.

The equivalence

of i) and ii) is due to Gluskin

[4], that of ii) and iv) to Wolfson

remaining part is new.

characterization

Jacobson

Another

([7], Chapter

II.3.5 THEOREM.

IV, Section

~

is an r-maximal

ii)

~

i__{sisomorphic

of these rings

[i]; the

csn be found in

16).

The following

i)

if it is

ri-semigroup.

conditions

on a ring

~

are equivalent.

simple atomic ring.

to the ring of all linear

transformations

of finite rank

on a left vector space. iii)

[P~

iv)

has a completely

~

implies

r-maximal

is a simple atomic ring in which

ideal

~r,~(L)

~

and

= 0

~ = C(~).

for a left ideal

L

L = ~.

PROOF. vector

0-simple

i) = ii).

spaces.

By 2.8, we have

~ ~ ~u(V)

for some pair

(U,V)

of dual

By 1.2.6, we have ~u(V) = is ~ ~(V) I a'V* ~ nat U]

which

implies

r-maximality

that of

ii) = iii). assume that 0-simple

gu(V)

is a right ideal of the simple ring

~, we must have We may take

R = $2,v,(V)

ri-semigroup

simple atomic ring

Hence

there exists

which

implies

~' ~ gUl( V' )

where

is a right ideal of

It is easy to verify S = ~°(I,G,A;P) is a nonempty

subset of

the columns of o (I , ,G,A;P I)

P

I, p1

indexed by

even though

V

that

S

K

KI =

(U~,V ')

S ~ $2,UI( V I ).

Thus

socle of a

of vector

spaces

~ ~ K ~ KI

right ideals of a Rees matrix

the semigroups

of the form

is the A x l l - m a t r i x

~°(I',G,A;p'),

obtained

from

I \ 1 I, where by abuse of notation,

~°(I1,~-,Iv1;p') I

and

$2,uJ(VI).

there may exist

$2,U,( V ) =

Hence by

of a completely

is the semigroup

k E A

such that

i' 6 I'. Hence

$(V).

N ~ ~(V).

is a left vector space,

a dual pair

that

that the nonzero

are precisely

Consequently

to a right ideal

By 3.2, we know

such that K'

~ = $(V), where

is isomorphic

S.

~'.

gu(V) = $(V).

~0

where J

11 ~

IV,

(I V ,~h-,Iv,;P).

since

P

semigroup where

11

by eliminating

we have written

pxil = 0

for all

87 If

dim V = i, then

which

forces

IV~

~2,uI(V~ )

K~ = ~2, uI(V t) = ~2,V ~,(V ~ ) ' isomorphism of ~ onto K ~. i j E I ~, then for any [i,?,X] ~ 0. 1.3.3.

U~ = ~u~

(~,a)

U~

a

iii) = iv). a right ideal of = ~2,v,(V). Let

L

L = ~v,(S)

completely

of

~2,v,(V),

where

be a left ideal of

transformation

a

onto

[i,~,~]K ~ = 0

Then

V

contradicting Hence

of 1.5,9 can be defined.

(&~,V ~) and

We conclude

and

that

K = S

that

~(~,a)l~ = @.

~2,V~,(V ~)

which

proving

of

~ = ~2,u(V)

and since

of the former implies

so that

for all

on

with

semigroup.

is not used.

K~

It then follows

implies

that

~2,u(V)

that

is

~2,u(V)

~ = ~(V).

Then the dual of 1.3.4 implies

that

V, and it is easy to see that =

= ~r,N(~v,(S))

s E S}.

It follows

(1)

(V)

~S~

such that

s(a*f) = (ss)f = Of = 0.

be an for all

that in the proof of 1.5.9 the im-

is both

~(V). S

0

pxi~ = 0

semigroup.

= ~(V)

S ~ = ~f E V* I sf = 0

linear

(&,V)

~ = Su(V).

for some subspace

and let

in the statement

the r-maximslity

~u(V)

~r,~(L)

b

@

0-simple

1

we have

ring

so that

is indeed a Rees matrix

and observe

under

We may take

But then

dim V >

is closed under addition

~

is also s division

dim V ~ = i

must have the same property,

and

K ~ = ~2,U ~(V ~) = ~2,V~,(V ~).

is an r-maximal

have

~, ~

isomorphism

the image of

Kl

~ E g-, we obtain

l i ~ I ~, ~ E ~] that

so

% E IV~

~°(I~,~I~-,Iv~;P I)

is a semilinear

that

and

and the functions

plicit hypothesis

Consequently

If for some

By the isomorphism

1.5.6 applies

ring,

But then

Suppose next that

j E It

Consequently

We let

is a division

to have only one element.

If

S # V, then there exists

Ss = 0. that

Then for any

a'V* ~ S ~

a nonzero

s E S, f ~ V*, we

and thus

0 # a E ~

(V). S~

Consequently, that

if

L = ~v,(V) iv) = i).

Further e E ~ which

let

A

by 1.6. is minimal

~r,~(L)

(U,V)

S = V

which

implies

= ~(V) = ~. Let

~

satisfying

be a minimal Hence

~I

iv) be a right ideal of a simple ring

right ideal of

= (e~)~ ~ c ~

that

~ = ~(V)

~

~.

= A

in view of its minimality

ring, and we may suppose 1.3.4, we have

= 0, then by (i) above we obtain

in

= ~u~(V )

for some subspace

Then

A = e~

so that ~.

Thus

where U

A~

U~ of

for some idempotent

is a right ideal of

~

~,

is a simple atomic

is a t-subspace

U~

~I.

of

V*.

By

and by 1.3.8, we have that

is a dual pair. The following argument

and notation

dual of 1.3.4 and (i) above, we obtain

is relative

to the dual pair

that if s subspace

S

of

V

(U,V). has

(V) = 0, then S = V. Let 0 # f E V*. Then S = Iv E V I vf = 0} S• subspace of V, and hence by the preceding statement, we must have ~ Consequently

By the

the property

is a proper (V) # 0. S±

88

s~ = [ u ~

ols~ = 0

Let

0 ~ g E S ~, then for every

and

g

sE

for all

v E V

are linearly independent,

s] ¢ O.

we have:

if

vg # 0

according to 1.2.3 applied to the dual pair

f = g~

for some

r ~ g-

which implies that

vf = O, then

then there exists

so that

f E U.

v ~ V

vg = 0.

such that

If

vf = 0

(V*,V), a contradiction.

Therefore

U = V*

and thus

f and Thus

U = U~

~ = ~.

The equivalence of ii) and iv) in the next corollary is due to W o l f s o n [i]; the rest is new. II.3.6 COROLLARY.

The following conditions on a ring

i)

~

is a primitive ring with an r-maximal socle.

ii)

~

is isomorphic

~

are equivalent.

to a ring of linear transformations on a left vector space

containing all transformations of finite rank. iii)

~I~

iv)

~

is a dense extension of an r-maximal completely 0-simple semigroup. is a primitive ring with a nonzero socle

a left ideal PROOF.

L

implies

i) = ii).

~u(V ) ~ ~i ~ £ u ( V ) But then

We know by 1.25 that

for some dual pair

~u(V) = ~(V)

ii) = iii).

and thus

We may take

is a dense extension of "ii) = iii)" in 3.5 that iii) = iv). L

a ~ ~r,N(L) that

~

and hence

~

ba # 0

and thus

for some ring

Since

~

~i

for

satisfying

is r-maximal, so is

for a left vector space

by 1.25.

~u(V).

V.

Hence

It was shown in the proof of

is a primitive ring with a nonzero socle ~r,N(L) = 0.

for some

Further let

b ~ L.

xa ¢ 0

~r,~(LA~)

and = 0.

0 ~ a E ~,

~.

Let

then

It follows from 1.25 and 1.3.6

and thus there exists

x = bach, we have

a ~ ~r,~(Ln~),

~r,~(L) = 0

is an r-maximal completely O-simple semigroup.

such that

is a regular ring,

Hence for

in which

~ £(V).

$(V) c ~ c ~(V)

$2,v,(V)

~

N ~ ~

(U,V).

~(V) ~ ~

~2,v,(V)

As before,

be a left ideal of

~

L ~ ~.

c E ~

x E L n ~

such that

ba = bacba.

which shows that

By 3.5 we must have

L N ~ = ~

so that

L ~ ~. iv) = i). ~r,~(L) = 0. 0 ~ a E ~, so

Let

L

As above

then

ab # 0

be a left ideal of the socle ~

is regular~ for some

Lab ~ 0, w h i c h implies that

hypothesis implies

that

L ~ ~.

and thus

b E ~ La # 0.

L

~

of

~

and suppose

is also a left ideal of

by 1.25 and 1.3.3. Consequently

Now 3.5 implies that

But then

~r,~(L) = 0 ~

that ~.

If

0 ~ ab ~

and the

is an r-maximal simple

ring.

The next result characterizes rings isomorphic space

V; other c h a r a c t e r i z a t i o n s

to

can be found in Leptin

£(V)

for some left vector

([2], Part I) and W o l f s o n [i].

89 II.3.7 COROLLARY.

The followin$ conditions on a ring

~

are equivalent.

i)

~

is a maximal primitive ring with an r-maximal socle.

ii)

~

is isomorphic

to the ring of all linear transformations on a left vector

space. iii)

~

is a maximal dense extension of an r-maximal completely O-simple semi-

group. is a semiprime ring having a minimal

iv)

right ideal

and is m a x i m a l among the semiprime rings

~r,~(A) = 0 ideal with

~'

A

such that

having

A

as a right

~ r , ~ ( A ) = 0. The equivalence of i), ii) and iii) is an immediate consequence of 1.27

PROOF. and 3.6.

i) = iv). such that

By 1.25,

~

~r,~(A) = O.

and having ideal of

A

is a semiprime ring and has a m i n i m a l right ideal

Let

be a semiprime ring containing

as a (automatically minimal)

~.

By 1.6, we have

e,e ~ E A, f E B.

Then

right ideal.

A = e~ = et~ I

A = ~

and 2.8, by 2.5 there exist y E ~,

~i

~ e~ ~ = e 1 ~ a,b E ~

and

= A, so that

such that

Let

B = ~

~

B

A

as a subring

be a minimal right

for some idempotents A = e~ j.

e = ab, f = be.

In view of 1.25 If

x E B

and

then xy = (fx)y = (baba)xy = b(ab)axy = fbe(axy) E fb ~e/~~ = fbA c f~ = B

which proves that right ideal of ~

of

~.

B

~.

is a right ideal of Consequently

But then

the socle

~

of

Since each minimal right ideal of

it follows that their sum r-maximality of

~

~, we have

Now the m a x i m a l i t y of iv) = ii).

~

Again

~u(V)

~

~ = ~J

since

~

~

where

are precisely

Ri = ~ [ i , ~ , X ]

must also be a minimal

is contained in the socle

~

and thus also of

satisfies

as a primitive ring with socle ~ ~ ~l

B

is a minimal right ideal of

is a right ideal of

~u(V) ~ ~' ~ £u(V)

dual vector spaces by 1.25 and 1.2.8. ideals of

~

~.

the conditions in 1.25. ~ = ~

implies

for some pair

and

~(V)

coincide. = 0

semiprime,

vxa = 0

Since

then

for all

the m a x i m a l i t y of

X E IV • ~,

£(V)

Ri

[i,~,~]a = 0

(vD,ui) J O, so that

of

I V ~ ~, X E I v ] .

and we have seen that

a ~ ~r,£(V)(Ri), such that

R i.

(U,V)

the sets

Thus the hypothesis on

for some

~ = ~.

We have seen in 1.3.11 that the minimal right

It follows easily that the sets of all minimal right ideals of the rings

~r,~(Ri)

~, By the

~

implies

satisfies

But then

~'

for all

a = 0 that

~ ~ g, ~ ~ I V . which

for

which shows that

3' = ~(V)

is semiprime and

the conditions

is a minimal right ideal of

(v ,ui)w(v a ) = O

we conclude

that

~u(V), ~

in 1.25, ~(V).

There exists ~ 9 0

it is

If ~ ~ Iv

yields

~r,~(v)(Ri) = O.

w h i c h finally yields

By

~ ~ ~(V).

90 The implication Wedderburn

"i) = ii)" in the next corollary

(or Wedderburn-Artiu)

theorem,

these rings was given by Gluskin [4],([7], Chapter

IV, Section

11.3.8 COROLLARY.

[i].

is known as the Second

A multiplicative

characterization

For further characterizations,

16) and Wolfson

The following

[ii.

conditions

~

are equivalent.

~

is a simple ring satisfying

ii)

~

is isomorphic

to a full matrix

iii)

~

is isomorphic

to the ring of all linear transformations

iv)

vector

d.c.c,

on a ring

i)

dimensional

of

see Jacobson

for right ideals.

ring

&n

over a division

ring

&.

on a finite

space.

~

is a prime

(respectively

~

is a simple

primitive)

ring satisfying

d.c.c,

for right

ideals. v) 0-simple

PROOF. minimal

ring a n d

~

is a maximal

dense extension

of a completely

semigroup. i) = v).

The second part of the hypothesis

right ideal, which

for some dual pair by 1.3.14 implies

that

But then 1.7.10 yields which is a completely extension

But then

dim V < ~. that

By 1.27,

which implies

dim V < ~.

that

d.c.c,

Since

ii) ~

We deduce

vector iii).

space

that

~ ~ ~u(V)

and thus

dense extension ~D~

has a

for right ideals which

= ~(V)

Consequently

~

~ ~ £(V). of

is a maximal

implies

that

is also simple, we must have

Consequently

for right ideals by 1.3.4.

iv) = iii).

gu(V)

is a maximal

~

~2,v,(V) dense

semigroup.

ideals and is both prime and primitive

dimensional

d.c.c,

the second part of the hypothesis

(U,V).

that

by 2.8 implies

satisfies

Consequently

semigroup.

0-simple

for some dual pair

satisfies

~u(V)

[k~(V) = g(V)

O-simple

of a completely

v) = iv).

together with simplicity

(U,V).

implies

nat U = V* Hence

~

~ ~ £u(V)

~u(V) = £u(V)

by 1o3.13 so that

£(V)

satisfies

d.c.c,

for right

~ ~ ~(V)

for a finite

by 1.25,

from 1.25 and 1.3.14

that

V.

This is a well-known

fact proved in elementary

texts on linear

algebra. iii) = i). Further

~(V)

fies d.c.c, Again

First note that

= gv,(V)

where

~(V)

is simple by 1.3.6 since

dim V < ~

so that 1.3.4 implies

~(V) = ~(V). that

~(V)

satis-

for right ideals. the hypothesis

corresponding

hypothesis

on right

ideals in i) and iv) can be substituted

on left ideals,

be taken on a left or a right vector

and in iii) the linear

space.

by the

transformations

can

91 11.3.9 EXERCISES. i)

Let

ideal of

~

~

be a primitive ring with a nonzero socle

~.

Show that a right

which contains s m i n i m a l left ideal must contain

A nonempty subset

A

for some n o n e m p t y subset

of a ring B

of

~;

~

~.

is called s left annihilator if

A = ~%,~(B)

a risht annihilator is defined dually.

It is

clear that a left (right) annihilator is a left (right) ideal. ii) ~(V)

Let

V

he a vector space.

coincide with right

Show that the principal right

(left) annihilators of

dimensional if and only if every right

£(V).

(left) ideal of

(left) ideals of

Deduce that £(V)

V

is a right

is finite (left)

annihilator. iii)

Let

~

be a regular ring with identity.

right (left) ideals of ~(V)

~

a sum of two right

is a principal

right

(left) annihilators

For the first statement, if A and B 2 e = e and b, respectively, consider

Show that s sum of two principsl

(left) ideal of

is s right

~.

Deduce that in

(left) annihilator.

(Hint:

are principal right ideals generated by e + (l-e)bx(l-e)

where

(l-e)h

= (l-e)bx(l-e)b.) iv)

Show that in any ring the intersection of any set of right

annihilators is a right

(left) annihilator.

of any set of principal right v)

Let

= ~(V).

V

Deduce

that in

(left) ideals is a principal

be a finite dimensional vector space, U

~(V)

(left) the intersection

right (left)

ideal.

be a subspace of

V*

and

Show that ~u(V) = ~r,$( [c ~ ~ I c*U = 0}).

vi)

Give an example of a ring

~

and two principal right ideals of

~

whose

~

whose

intersection is not a principal right ideal. vii)

Give an example of a ring

and two principal right ideals of

sum is not a principal right ideal.

II~4

SEMIPRIME

RINGS

We are interested here in semiprime rings with a nonzero socle. us a general result,

This will give

several special cases of which will be treated in greater

detail in the two succeeding sections.

Some new notions will come in handy

(recall

1.15). 11.4.1 DEFINITION. every

i ~ I, we have

Ai, i E I, if =

~ Ai . iEl

~ =

A family

~AiJiE I

A. • ~ A. = 0. I j#i j

~ A. iEl l

and the family

of subrings of

A ring

~

~Ai}i61

~

is independent

if for

is a direct sum of its subrings is independent,

to be denoted by

92

11.4.2

DEFINITION.

subsemigroups denoted

by

Such

A semigroup

S , ~ ~ A,

S =

~ ~ o@A

a "sum"

S

if

S =

as follows.

zeroes

0~.

Let

Let

S =

with

zero

and

S S~ = S

is an orthogonal ~ S~ = 0

sum of its

if

~ # ~,

to be

•

of semigroups

proceed

S

U S

may be termed

~S~c~A_

[ U

(S

'internal".

Externally,

be a family of pairwise

\03]

U 0, where

0

disjoint

is an extra

we may

semigroups

symbol,

with

with multi-

~A plication

a*b

in all other groups

= ab

a,b ~ S for some ~ ~ A and ab # 0 , and ct c~ In this case also we say that S is an ortho$onal

cases.

S c~, ~ 6

theorem,

if

A, with

see D i e u d o n n @

the same notation. [3] and J a c o b s o n

For

the origin

([7], C h a p t e r

a*b

= 0

su.__mmof semi-

of a part of the next

IV, Section

3),

the rest is

new. II.4.3 minimal

THEOREM.

risht

Let

ideals

~

be a semiprime

assumed R

=

U

nonempty. Ri,

~

=

be the socles

of

~lR

i T j <= R. --~ R. z j and

Then each of all all

~

~

.

%.

minimal

=

~

~

= C(~

yield T

that

~.

ring by

elE

each of these

left

and

steps,

~

If implies

which ~

that

~ ~

¢ 0, then

and

Hence

this

follows 1.16

R

~

by:

Let

J = I/~

is an ortho$onal ~

sum

is a direct

sum of

definitions

with

~

immediately

that

ideals

.

Consequently since

~.

these

are

contains

is ~ - i s o m o r p h i c

~

R

.

definitions ~

agree ~

ideals of

as

e, e ~.

and by 2.5

each

that

a minimal

isomorphic

in some

Consequently it follows ~

the

idempotents

of

are contained

~

and

ideals

the c o r r e s p o n d i n g

~

from

the left socle of left

for some

right

they

(and of

intersection in

~

relation.

and

L ~ = ~e ~

that

is an ideal of

ideal

~,

are minimal

~, R, % ,

that both

a relation

the analo$ous

are minimal

~X~

~R

of a sum of subrings,

then implies

every right

and

L~

we conclude

ideals of

and hence

of all

, ~.

L = ~e,

the same

J).

of

relation

and

e~

define

(~ E

of

ideal

We know by

L

as ~ - m o d u l e s .

By the very d e f i n i t i o n

= C(R),

ideal

~, ~, ~

1.6 we have

left ideals yield

1.13),

If

I

R

), ~ = C(£),

isomorphism.

then by

a union of m i n i m a l

~

atomic

the same

the socle of

On

is an equivalence

=

is an equivalence

they are isomorphic

reversing

~

0-simple

is a simple

of a module

By 1.14, we have

minimal

R.

is a completely

left ~ - m o d u l e s ,

[~.

=

Each

That

agrees w i t h

respectively. Then

Furthermore,

definition

~,

U

~Ri}i~ I

i~l

as ~ - m o d u l e s .

left ideals

PROOF.

that

and

the set

Ri

~

i~l

rin$ with

Let

By using

is also

in a semiprime is an ideal of

~

= C(~

),

~.

right

to any minimal

ideal which right

ideal

in

93 R~,

so we must

since both R

have

~

R

and

= 9~.

R~

It follows

are ideals

of

that

R(~

~ 9

[F~i, so that

~ ~

R

= 0

if

~ #

is an orthogonal

sum of all

•

We prove next ideal of

~

a~ = R i

that

contained

and thus

Let

I

~

a~

that

idempotent

minimal

left

for every

t 6 R

there exists

iv)"

conclude

I Q ~

that

Let

y ~ Rj

J

are N - i s o m o r p h i c , =

that

quently

~_ I.

~

~

shown above

nonzero

R i = a~ ideal of

.

~

is also

in

right

again by

.

in 2.8, we

such

let

the union of

by idempotents.

~

through w i t h obvious

Hence ~

of "v) = iv)"

by [.6 are g e n e r a t e d

and

let

ideals e.

that

Hence

t = te.

modifications,

Ri

x~ = e.~l

that

I Q ~

idempotents,

Rj•

in

~

of •

implies

that

~ 0

The

and we

x £ R.

By 1 . 6 ,

~.

of

a,b ~ ~

Ri = e i ~ R.

and

such

that

for some

J, since

and

and R.

z ~ ~

b,zay ~ ~ .

It

Conse-

ideals.

which

, it follows

Then

e i = xz

= bx(zay) £ nonzero

.

By hypothesis

the existence

= beiaY

~

and e.

has no proper

generates

0 # x 6 J, y ~ R

and

2.5 implies

Further,

~

that

that

e

be a m i n i m a l

right

In the proof we have

an idempotent

idempotents

and

Since

contains

which

y = ejy = b ( a b ) a y

J = ~

It was ~

of

which by

ab, e.j = ha.

follows

.

Ri

# 0.

be an ideal

for some

~

Let

By the dual of i.ii, we have

implies

is a m i n i m a l

above,

in 2.8 now goes

f o r some m i n i m a l r i g h t

R. = e.~

ei

ideals,

which R.

ideal of

ring.

0 # a ~ R i.

a~ = R

the first p a r a g r a p h

proof of "v) =

atomic

and let ~

we conclude

be a nonzero

From

is a simple

in

Ri ~

the dual of i.ii,

= ~ .

~

now yields

that

it must be simple.

I ~ ~

I = ~.

= ~

Finally,

Therefore

~

and

thus

since

~

is a simple

atomic

ring. We know a minima[ then

A = e~

and

A

~

socle of

.

%(

=

~

~(%)

we deduce ~ ~j

consider

seen above %)

= 0

is equal ~

A

~

atomic

~u(V)). for

in v i e w

to zero

since

~

~

is,

~

~

But

right

of

%,

~

socle

= 0 ~ a = 0

is a simple

atomic

[~c~A

sum of all

~ of

in

that

is completely

then

in

ideal

A~ = ~ ~ c

of its m i n i m a l i t y ideals

R

the family

contained

then

the semigroup

~

is a direct

of

is a m i n i m a l

~ ~ 8, we have ~

that

right

ring,

a E ~

that

ideal

A

by 1.6.

Therefore

and if

= 0,

if

e E ~

~, m i n i m a l

that

right

Conversely,

idempotent

ideal of

, it follows ~

.

But in a simple

(e.g.

We have Hence

for some

~

is the union of all m i n i m a l ~

0-simple

that every m i n i m a l

ideal of

is a right

quently

But

from above

right

~

is,

.

.

,

= A

Conse-

the semigroup

is completely O-simple. ~~ ~

so that so that ring.

=

0.

a t ~£(%).

Thus

is independent. ~

e~

is ~

a = 0 Since

and

94 11.4.4 COROLLARY.

A semiprime ~

i_~f [i~

has a completely 0-simple

if

is a minimal right ideal of

Ri

product, and PROOF. ideal of Also

~

= (~)Ri,

~

contained in

~

has a minimal

right ideal if and only

In fac___~t, in the n o t a t i o n of th__~etheorem, contained in

~ , then

~

= ~Ri,

the ring

the semigroup ~roduct.

The first statement

(fi~)Ri

~

ideal.

~

is an ideal of

follows

from 4.3 and 1.13.

and thus must coincide with ~[~

contained

in

R

Further, ~R. i ~ since ~

is an is simple.

so it must coincide with

since a completely 0-simple semigroup has no proper nonzero ideals.

II.4.5 EXERCISES. i) that A

Let

A

AB = 0

into

B

and

if

B

be R - i s o m o r p h i c minimal right ideals of a ring

B2 = 0

and

AB = A

A

of

A

into

A 2 # 0, find

~,

a sum of all minimal right ideals

of

~

~; a homogeneous component of the left socle is defined analogously.

of

3. iii)

A

If

of

ii)

to

is called a homogeneous

component of the socle

Show that a homogeneous component of the socle of a ring

Show that the nonzero socle

homogeneous components.

~

of a ring

~

~

is an ideal

is a direct sum of its

Also prove that each homogeneous component of

direct sum of pairwise ~ - i s o m o r p h i c minimal second statement,

consider independent

Show

B.

is a minimal right ideal of a ring

which are ~ - i s o m o r p h i c

~.

B 2 # 0, and that an ~ - h o m o m o r p h i s m of

is either a zero ~ - h o m o m o r p h i s m or an ~ - i s o m o r p h i s m .

the form of all ~ - h o m o m o r p h i s m s If

if

right ideals of

3.

~

(Hint:

is a For the

families of minimal right ideals and use

Zorn~s lemma.) iv)

Let

F

be a homogeneous component of the socle of a ring

be a sum of all nilpotent minimal right ideals of N

is a nilpotent ideal of v)

~

and that

~

contained in

~, F.

and let

N

Show that

N = F N ~r,~(F).

Give an example of a ring which coincides with a homogeneous

component of

its socle and contains both idempotent and nilpotent minimal right ideals. vi)

Prove that if a homogeneous component

semiprime, socle of

then

F

F

of the socle of a ring

is a simple atomic ring and a h o m o g e n e o u s

~

is

component of the left

~.

For general references on this subject, 2), Jacobson ([7], Chapter IV, Sections

see Behrens

1,2,3).

([i], Chapter IV, Sections

i,

95

II.5

SEMIPRIME RINGS E S S E N T I A L EXTENSIONS OF THEIR SOCLES

The principal result here is a collection of c h a r a c t e r i z a t i o n s of semiprime rings which are essential extensions of their nonzero socles.

As a preparation,

we first study the relationship of prime and semiprime ideals in any ring. element

a

of s ring

11.5.1 LEMMA. i)

I

ii)

I

If

A

and

B~

PROOF.

I, then either B

i) = ii).

a E I

and

or

B

there exists we obtain

If

n ~ p.aq. In i= I i 1

such that

of

generated by

a.

~

are equivalent.

b ~ I.

~

such that

(a) c

ABc

I, then either

I

or

(b) ~ I

so that

I

a E ~,

is an integer,

~

such that

a ~ I.

(a)(b) c AB + ~ A B ~

For any

and thus

r,s Pi qi E ~}. ' '

ABc

I.

(i)

Suppose that

A ~

I; then

b E B, using (i) and the hypothesis,

b E I

since

a ~ l.

Hence

B G I.

Trivial. Every semiprime ideal of s rin$

~

is the intersection o f prime

~.

PROOF.

Let

I; then

I

which implies

r ~ I, then We let

be a semiprime

(~r~) 2 c ~(r!Rr)~ ~

(r) 3 c ~riR ~ I if

or

(a)(b) ~ I, then either

be right ideals of a E A

11.5.2 LEMMA.

riRr ~

1

~

b 6 I, First note that for any

iii) = i).

ideals of

a ~ I

are right ideals of

(a) = I n a + r a + a s + A

ideal of

I.

ii) = iii).

Let

the principal

The followin$ conditions on an ideal

(a)(b) ~

or

either

let (a) denote

is a prime ideal.

If

iii) Ac

~,

For an

rxr ~ I

r E ~\ I

I

that

for some

ideal of where r E

I.

2. ~r~

Let

such that

r 2 = rlxr I ~ I.

Xl,X2,...

such that

Starting with

p

such that

We can inductively define sequences

rn+ I = rnXnr n ~ I.

Let

M = irl,r2,...}.

r.j[xj(rj_iXj_l) ... (ri+iXi+l)(rixi)]r i = rj+ I E M. c,d E M,

then

~r~ ~

I.

cxd E M

for some

r ~ P

r I = r, there exists

r i[(xiri)(xi+Iri+l) ... (xj_irj_l)xj]rj = r j+ I E M,

if

so

But then

We will use this in the following form:

easily o b t a i n

Consequently,

and suppose that

x C ~.

and wish to find a prime ideal

P ~ I, w h i c h will prove the lemma.

r 6 ~

is an ideal,

x E ~.

rl,r2,.., For

and

xI E and

i ~ j, we

96 Now

let

~

for w h i c h standard

be

Zorn's

lemma

to show

that

P

P + (a) D

P

which

there exists that

then

the p a r t i a l l y

J fl M = ~

and

argument

is in fact

r ~ P

and

~

P

x E ~.

If

d 6

P

ideals I E ~.

P.

In order

a,b ~ P.

and hence

We have

Consequently

is a prime

ideal.

(a)(b) ~ Since

and is also denoted its ~-th

Any subring sum

consisting ponents

Let

[~#o~A_

(also

of

be a family

~ ~

of rings.

On the C a r t e s i a n

PROOF.

consists then

Thus

subdirect

element

homomorphism,

projection

of the rings

defined

earlier

~

.

of

~

to be d e n o t e d

homomorphisms

are onto

The subdirect

by

is a

sum of

the same way.

is sometimes

and external

An ideal ideals.

I

direct

called

sums

"internal

is v e r y

direct

close

and the

of

~

i_~s semiprime

if and only i f

I

i__~s

A ring

~

is semiprime

if and only

~

is a

of the first

~ I

statement

for some prime

It follows

that

is the content

verification ideals

the m a p p i n g

I r ~

of 5.2;

rings

~

the proof of

and is omitted. of

~

by

(r + l ) a 6 A

, ~ ~ A, we can identify

~

with

If

~

is semiprime.

is

statement

is an i s o m o r p h i s m are evidently if

~

a subring

~ • If now I is a n i l p o t e n t ideal of ~, then In is a n i l p o t e n t o~A ~ ~ , which implies that I ~ = 0 . Since ~ C A is arbitrary, we obtain Consequently

~

the first

~ (~/I) and the p r o j e c t i o n h o m o m o r p h i s m s of the image o~A ~ is a subdirect sum of prime rings ~ / I • Conversely,

sum of prime

if

rings.

of a s t r a i g h t f o r w a r d

0 =

of the proposition.

onto.

product)

of internal

of ~rime

Necessity

sulficiency

into

all of whose

to every

of the rings

is rarely made.

sum of prime

~

projection

"sums" will be labeled

sum" of rings

the i n t e r s e c t i o n

of

The m a p p i n g which

The resulting

~ ~ with only a finite number of nonzero como~A ~ (also discrete) direct sum of the rings ~ . Any ring

II.5.4 PROPOSITION.

semiprime,

we have

of

to any of these

distinction

•

is the ~-th

subdirect

The r e l a t i o n s h i p

subdirect

~ ~

component

is an external

A "direct sum".

~

by

of all elements

isomorphic

which

P E ~

~ ~ define the a d d i t i o n and m u l t i p l i c a t i o n coordinatewise. o6A ~ structure is a ring called a complete direct sum (also direct product)

subdirect

P

I.

11.5.3 DEFINITION.

.

seen above

then

product

associates

of A

Then

( P + (a)) n M # ~

p,

J

so

p

P N M = @.

that

element

that

( P + (b)) ~ M.

(a)(b) c

( P + ( a ) ) ( p + (b)) c

implies

that

of all

I N M = ~

has a maximal

implies

contradicting

and 5.1

since

ideal, we suppose

and s i m i l a r l y

for some

P P, M

P ~

that

(under inclusion)

~ # ~

(p + (a)) N M

c ~

by c o n t r s p o s i t i v e

shows

a prime

set

Then

of

cxd E M

exd E

J.

by m a x i m a l i t y

cxd E and thus

ordered

I ~

is a of

ideal of I = 0.

97 II.5.5 DEFINITION. product

II S

~A

Let

~S }¢~ A

be a family of semigroups.

The C a r t e s i a n

together w i t h c o o r d i n a t e w i s e m u l t i p l i c a t i o n is a direct product of

(7

semigroups

S , ~ ~ A, and is also denoted by

~ S •

o~A ~ II.5 • 6 LEMMA.

Let

~ ~ S , where for each 0~A ~

S =

~:(k'P) -- (~'lS '~IS )c~:A is a n

isomorphism of

Then the m a p p i n g

((k,p) ~ fl(S))

~I ~(S(~). o~A PROOF. Let X be a left translation of S, a E S , and suppose that ha ~ 0. 2 Then a = xy since S = S so that Xa = X(xy) = (Xx)y # 0 which implies that ~y Xx ~ S • Hence ~ maps S~ into itself. Consequently ~ maps [~(S) into II ~(S(7).

fl(S)

(7, S 2 = S • ~

onto

V e r i f i c a t i o n of the remaining properties of

~

is left as an exercise.

~A II.5.7 DEFINITION. 0-simple semigroups

A semigroup which is an orthogonal sum of completely

is a primitive r e g u l a r semigroup.

It is clear that a primitive regular semigroup is indeed regular, more that each of its nonzero idempotents of s primitive regular semigroup.

is primitive;

and further-

this is the usual d e f i n i t i o n

The two definitions are equivalent,

see c l i f f o r d

and preston ([i], Chapter 6, Section 5). II.5.8 LEMMA.

--Let ~R =

~ ~ c~A

minimal right ideals. if for some

only

A subset

(7 E A

Necessity.

same as in 5.4. some

(7 E A.

0

is a semi~rime ring having ct

R

of

~

E R,

a~

First note that

Now let

Let

~

is a minimal right ideal of

and some minimal right ideal

R = ~a E ~ l a~ PROOF.

, w h e r e each ct

R

= 0 ~

if

R

of

~

~

if and

, we have

~ # ~}.

is a semiprime ring; the proof is the

be a minimal right ideal of

3,

then

R~

~ 0

for

E R~ ; then there exists 0 ~ a E R such that an = a . (7 cy ~ c~ (7 By the dual of i.ii, we have s~ = R so that s ~ = R~ w h i c h again by the dual of i.ii implies that

@ a

R~

is a minimal

D = [r E R I r ~ If

r E D

and

s E R, then

a right ideal of (recall that a ~ R, we have we obtain in

R

~.

~

r

0 # ab ~ D.

r~

~ a

and

.

Let

~ # (7}.

= (r~)(s~) E R~

~

= 0 b

E ~

and

rs E R, so that

be such that

a b

is semiprime so it has trivial right annihilator).

Let

b E ~

Consequently

w h i c h implies that

=

for all

(rs)~ 0

an~ = a ; letting

Sufficiency. and

Let

= 0

right ideal of

D

be such that

b~

D

is

# 0

Then for some

= b , b~

= 0~

is a nonzero right ideal of

~

if

~ # ~,

contained

R = D.

s,r E R

and suppose that

a # 0.

~ R(7, which by the dual of i.ii implies that

Then

0

# s

r~ = a~x~

= an

E R

for some

98 x~ ~ ~.

Now letting

obtain

ax = r.

x E ~

be such that

Consequently

dual of i.ii implies

that

We are now ready

R

a~ = R

x~

= x~

for every

and

x~ 6 = 06

0 ~ a ~ R

is a minimal right ideal of

if

6 # ~, we

w h i c h again by the

~.

to prove the m a i n result of this section.

This result is new.

Another c h a r a c t e r i z a t i o n of the rings in the next theorem was given by Jaffard 11.5.9 THEOREM. i)

~

ii)

~

~

~

are e@uivalent.

i~s ~ semiprime ring essential extension of its nonzero socle.

Ever~ nonzero ideal of

iii) each

The following conditions on a ring

[ii.

i_~s isomorphic

~

contains an idempotent minimal right ideal.

to a subrin$ o f

~ ~ containing the socle thereof, where ~A ~ -is - _ a prime ring with minimal right ideals.

iv)

~

is a dense extension of a primitive regular semigroup.

v)

~

has a primitive regular ideal having nonzero intersection with every

nonzero ideal of PROOF.

~.

i) = ii).

Since

right ideal is idempotent. we have then

~ N I ~ 0

since

~

a = a l + a 2 + ... + a

components of b I E %1

~

is semiprime,

Let

~

such that

1.6 implies

the socle of

ai E ~

, where

n I

alb I @ 0; on the other hand

and thus

For a nonzero ideal

~i

~

~

~.

Let

0 # a E ~ A I,

are some homogeneous

aib I = 0

I.

But then

I

for

1 < i < n

Hence

since

0 ~ ab l

contains an idempotent minimal

right ideal. ii) = iii). hence of

~

~.

The hypothesis

is semiprime. If

A

for every

By 5.2,

implies that

~

the ideal

is the intersection of prime ideals

0

cannot have nilpotent

is an index set of homogeneous components of the socle

~ E A

we can choose a prime ideal

I

such that

I

ideals and

~

N ~

of

= 0

~,

since

is simple. Assume that

N I , then (a) is a nonzero ideal and hence by o~A ~ hypothesis contains a m i n i m a l right ideal R of ~. Hence R ~ (a) c N I

0 ~ a ~

--

thus 66

R c

I

for all

and hence

But

R

Consequently

N I ~ A

-

is a prime ring and the m a p p i n g

onto a subdirect sum of rings

-

~

.

o~A

and ~

is contained in some homogeneous component

R c 6 6 n 16 = 0, a contradiction. -

~ = ~/I~

~ E A.

I,

oi ~i is a simple atomic ring and thus there exists

~

distinct homogeneous components annihilate each other. = alb I E ~ i

that every minimal

~.

is an essential extension of

for some

n By 4.3, each

6.

denote

~:r ~ (r + I )o~ A

= 0.

Hence

~

is an isomorphism of

99 The canonical ~

~ I~ = 0~.

an ideal of

homomorphism

Hence ~

.

~

contained

~e

ideal,

~e"

Next let @ A

M

Since

) ~

since

~e

and thus

that every ~e

is

is a prime ring, its socle is contained that

~e

right ideal of

right ideal

M = [(a

@e

is the socle of

~.

Hence

~

right ideals.

be a minimal

and a minimal

right ideal of

~

and we conclude

is a prime ring with minimal

on

of a simple atomic ring, it follows

is a minimal

in the socle of

is one-to-one

so that the latter is a simple atomic ring and also

From the properties

minimal right ideal of

in every nonzero

~e :N ~ ~e = ~/I~

~ ~e~ e

M~

~ ~

of

N ~ . Then by 5.8, o~A ~ such that

~

I a e E Me,

a

= 0

if

there exists

~ # e}.

(i)

o6A ~ Furthermore,

we must have

M~ ~ ~e~ e

and thus

~ ~ . o~A ~ iii) = iv). We may suppose that

M ~ ~.

Consequently

~

contains

the socle of

~ ~

R

with property

-- c ~ A

are pairwise disjoint. group socle

~

of

We know by 1.25 that

~

.

a -f'r(:y

of

.el~

U ~',

~ =

~

M

the hypothesis Consequently ! b E ~e with ab = O.

~

®

, .

c~

R ei

into

~e'

given in (I) above is a minimal implies that

~' ~ ~

M c-- ~.

and thus

~ ~ ~, then

It follows

is a primitive

that

(ab)~ ~'

regular

is an isomorphism

(a~)~A

of

~

It follows

~ =

(a~7)(b~)

semigroup.

that

G(~

).

= k a I~ ' PI~

a' E ~ ,

we obtain

~np

maps

O-simple.

If

for all

[1 ~ by 5.8 and c~ o~A 9'~ onto ~ . a E ~

and

W E A, and thus ~

and therefore

t h e mapping

(a ~ ~ ) Hence by 5.6, we have that

~ (k a I~ ' Pa I~ ) ~ ( k , p )

kl~

is a minimal right ideal

sum of semigroups

By 1 . 7 . 5 ,

)

Me

right ideal of

= 0

is an orthogonal

into

If

is completely

a ~ (kal ~ , pal~

where

~

c~A

is an isomorphism of

IRe, then

is a dense extension of the semi-

C~' a'r[ ~,

o~A ~ Then

BIR

'01 c~

Define

(Y ~' =

iii), and that

~

((a)~A

= O a I~ , are isomorphisms

E ~),

mapping

~A For

I

a' = (a~)~A

~ (Xa, la , Pa':~

) ~ (X:'~a)

E :(~)

(2>

i00 where

a = aJ~

a E ~, then istence of

and

~(~)

for some

a~ E ~

for which

mapped onto

(Xa,Pa).

also maps

is the inner part of

a ~ ~

~

onto

~ E A. a'~

It follows ~(~)

If

I

a

and

ar~

= 0~

which by 1.7.5 implies

that

~

~I~

by:

is an ideal of

a o b

that

[PS~ such tLat

if and only if

is the equality relation.

be the equality relation which v) = i).

Let

semigroup socle of ~.

If

R

regular, Hence

= C(R)

J ~ ~' = 0 Thus

~

~

But

~ ~

Let:

I

~.

R

implies

that

~

must

J = I U ~R

that

that I

Then

I

where of

~

R~

is

rxa E ~' N R = 0.

~

J = 0.

~ = ~.

I = 0.

be the

is an ideal of

s E ~, r E R, and since

is an ideal of

If

~

be a minimal right ideal of

implies that

which proves If

o

a = b, has the property

a = axa = s(rxa)

The ideal

and the hypothesis implies

is a nonzero ideal of .

or

I = (~k~)(~R).

for some

w h i c h by hypothesis

extension of its nonzero socle

Therefore

by 4.3.

such that

~.

is a

R, then the congruence

a,b ~ I

consider

a = sr

R c F~

~

1 = 0.

~ ~ ~,

~ I = 0.

is the socle of

~I ~ R = 0

I N ~ = 0, where

Then

x E ~J

is

~(~)

is a dense extension of

~.

and thus

contradiction.

into

be the primitive regular ~deal in v), and let

there exists

property

~

~

~

a E ~' N I, then

a = 0

~ # ~; in (2), a ~

But then the hypothesis

forces

is not contained in

and if

if

if

implies the ex-

~.

dense extension of the primitive regular semigroup defined on

On the other hand,

the hypothesis

that the isomorphism w h i c h maps

the primitive regular semigroup iv) ~ v).

~(~).

As above,

then has the

But then

R = 0, a

By 4.3, we have that such that

Consequently

is a nonzero ideal of

~ [~ I = 0, then B ~,

is an essential then

~I ~

~, and thus must contain a completely 0-simple component

always contains a nonzero idempotent

so that

I

cannot be nilpotent.

is a semiprime ring.

Note that the rings minimal right ideals. according to 5.8,

for each

a prime ring by 1.25.

~

in the theorem are subdirect sums of prime rings with

For if

~

is a subring of

~ ~ containing its socle, then (~A ~ must contain the socle of ~ and is thus

~ ~ A, ~

It is clear that the rings in the theorem do not represent

the general case of subdirect sums of prime rings with minimal right ideals.

In

order to deduce a consequence of the theorem we first need a ring analogue of 5.6. II.5.10 LEMMA.

Let

~ = ¢~A~'

where for each

~ E A, ~ 2 = ~

mapping ~:(k'P) ~ (~I~ ' PI~ )o~A is an isomorphism of

~(~)

onto

~ ~(~ QEA

).

((k,0) E Q ( ~ ) )

.

Then the

i01

PROOF.

Let )~ be a left translation of n a = i~__ixiYi for some xiY i E [Rc~ since

Then

3, a ( ~c~' and suppose that 3 2~ = ~

ka # 0.

so that

n 0 ~ ~a = ~(i~__ixiYi) = i=l~(Xxi)Y i. Since

~c~

ha ~ ~

.

is an ideal of

3

and

This shows that

~

maps

valid for r i g h t I~ ~(~ ).

translations

That

c~A ~ of a direct

~

-lY- ~ 51(~, we must have 3

which

into itself.

implies

that

indeed

maps

f~(~)

into

sum a n d

is

left

as

an exercise.

The following conditions on a ring

~

are equivalent.

i)

~

i__s_s~ semiprime ring maximal essential extension of its nonzero socle.

ii)

3

is a complete direct sum of maximal primitive rings.

iii)

F~

is a maximal dense extension of a primitive regular semigroup.

PROOF.

i) = ii).

Since

~

is semiprime, it has a trivial annihilator and

hence the ring analogue of 1.7.5 implies that

~u

and thus

has the remaining properties follows easily from the definition

II.5.11 COROLLARY.

3.

~

_(kxi)Yi ~ ~c~

Similarly the same result is

By 4.3, we have (A ,Vc~)

~ ~ ~(~), where

~

is the socle of

~ =

+G ~ , where each ~ is isomorphic to a ring of the form c~A c~ c~ by 2.8 so that ~2c~ = ~c~" Hence 5.10 yields [~(~) ~ II ~(~c~ )" By

c~A 1.7.13, we have

~(~c~ ) ~ £U (Ac~'Vc~)' where

ii) = iii). pairs

£U (4~'Vc~)

By 1.27 and 1.7.13, we have

(U ,4c,Vc~).

F~ ~

But by 1.7.10, gU (4(,V(~)

completely O-simple semigroup

is a maximal primitive ring.

~ gU (~c~'Vc~) for some dual ~fA is a maximal dense extension of the

~2,U (Ac~'V~)" Now using 1.7.5 and 5.6, we obtain

-~ ~ g,, (4 ,v ) ~ E P(~o , (4 ,v )) ~ Q ( ~ ® ~2,u c~A Uc~ c~ c~ o~A ~,u ~ a c~A where

¢~A~ ~) 52'U (A ,V )

(4 ,v ))

is a primitive regular semigroup which again by 1.7.5

implies that iii) holds. iii) = i). socle.

containing that

By 5.9, ~

is a semiprime ring essential extension of its nonzero

By the ring analogue of 1.7.5, we may identify

~

~(~), where

~

is the socle of

~.

is a dense extension of the semigroup socle

regular semigroup.

On the other hand

~ =

~ %'

%

~ ~2,U (4 ,V )

as

before.

$2,U(A,V)

~(~)

~

of

3

which is a primitive

~ O ~ , a~..A In the proof of 1.7.13, we have seen that the

correspondence which to each bitranslation of to

with a subring of

~ ~U (4 ,V ), ~ =

c~A

~

~

It follows from the proof of 5.9

is an isomorphism of

~

~U(4,V)

~I'~(~U(4,V)) onto

associates its restriction ~(~2,U(4,V))

so the same

102 must hold for

~

and

~ .

5.6, by taking direct dense extension

of

Hence

products,

~(~),

identified

extension of

~.

by hypothesis

of maximality

ring analogue

of 1.7.5 yields

zero socle

~f~(~) ~ ( ~

yields

We have already

above with

implies

that ~

which

~ ~(~).

seen that

that

)

~(~)

~, so

~

~

together with 5.10 and

By 1.7.5, ~(~) ~(~)

is a maximal

is a maximal

is a dense extension

= ~(~)

is a maximal

and thus essential

of

dense

~

~ = ~(~).

which The

extension of its non-

~.

II.5.12 EXERCISES. i)

Establish

ii)

the relationship

Show that in 5.9,

iii)

this for semiprime

implies

1

and on subdirect

~

ideals,

([i], Chapter

SEMIPRIME

several

(a)r

the principal

the last two references right ideals.

II.6.1 THEOREM.

thereof.

contain

further

The followin$ conditions

is a semiprime

rin$ satisfying

i) = ii).

rings.

Let

of

Then

~.

(al) r ~

aI = Xll+Xl2+

from 2.8 and 1.3.6 that each such that

Xli = xliei,

It follows

that

resular

ideal

By 4.3, we have

(a2) r ~

...

generated

~

~ =

a by

of a ring a.

~,

The equivs-

[i] and also to F. Sz~sz

information

on rings satisfying

is new. ~

ar___~eequivalent.

for principal

such that G ~

for some

right ideals.

~ = C(~).

, where

~

are simple atomic

chain of principal

right ideals

Xli E % i , 1 < i < n.

is a regular ring.

and thus letting

atomic rings

rinss.

be a descending

... + X l n ~

~

on a rin$

d.c.c

is a direct sum of simple atomic has a primitive

([i], Sections

2).

For an element

The rest of the theorem

ii)

see McCoy

of semiprime

is due to Faith

atomic ring.

PROOF.

an analogue of

ATOMIC RINGS

right ideal of

is a semiprime

iv)

Establish

ill, Section

characterizations

i)

iii)

~".

if and only if for sny

on prime and semiprime sums, Behreus

fence of ii) and iii) in the next theorem [1],[2];

is prime

b E I.

first establish

d.c.c f o r p r i n c i p a l

direct sums.

[g~" may be replaced by "ideal of

a ~ I

and then will deduce some consequences we denote by

snd internal

or

II.6 We will

of

of a ring

that either

external

ideals.

For more information 18-20),

part v), "ideal

Show that an ideal

a,b ~ ~, ~a~b ~ I

between

Hence

there exists

e = e l + e 2 + ... + e n ,

we obtain

We know e. E s I = ale.

103

(al) r = al~ = ( X l l + X l 2 + ... + X l n ) ~ +...+ = Xll%l Xln~ n Xll~+''" =

+Xln~

= (Xll) r + ... + (Xln) r. Since

for any

i

we have

(ai) r ~

(al)r,

this discussion

is valid

for any

a i = Xil + ... + X i n , and we obtain (Xll) r + (x12) r + ... + (Xln) r ~ Since

n ~ ~ i=l ~i

the sum

is direct

each of these

d.c.c,

(xji)r ~ ~ ,

(x2i)r ~

is a principal

for principals

right

...

right

ideals.

for

that

(am) r = (am+l) r = ...

i < i < n,

ideal of

~i

By 2.8

Hence we can find

(Xmi)r = (Xm+l,i) r = ... But then

we conclude

....

i

(Xli)r ~ where

and

(x21) r + (x22) r + ... + (X2n) r ~

and thus

for ~

m

satisfies

each

such that

1 < i < n.

satisfies

d.c.c,

for principal

right

ideals. ii) = iii). minimal

We have

principal

right

seen in the proof of "iii) = ii)" in 2.8 that every

ideal

is a minimal

right

ideal.

Since

by 1.6, every minimal

right

ideal is of the form

e/E

Hence

right

ideal of

a minimal

every principal

e~.

Let

a E ~

(a)r ~ e ~

and consider

(a)r.

for some idempotent

eI

~

contains

If

(s) r

~

for some idempotent right

is not a minimal

such that

el~

is semiprime, e.

ideal of the form right

is a minimal

ideal,

right

then

ideal.

Since a = e l a + (a - els) 6 e l ~ + (a - ela)r, we obtain

(a)r ~ e l ~ + (a- ela)r.

Conversely,

for any integer

n ( a - ela ) = n a - e l ( n s ) ~_ ( a ) r + e l ~ and for

s 6 ~, we have

el~+ (a-ela)r If

(al) r

(a)r.

e~

Hence

letting

right

contained

a 2 = a I -e2al,

in

we have

principal

(al)r, where

(al) r = e ~ +

ideals.

as a sum of elements

Thus

e2

right

ideals

that

~

is a direct

ideal of We deduce

r ~

that

(a2) r.

of steps because

e.~.

+en~

of the d.c.c,

so that

Therefore

~

a

sum of simple

atomic

for

can be written

is an atomic

i

which by 4.3 implies

implies

Hence

(a)r = e l N + e ~ + . . .

of minimal

right

is an idempotent.

(a2) r.

~ e~+(a2)

which

(a)r = e l ~ + ( a l ) r.

we can find a minimal

must end after a finite number

right

(a)r

a I = a - e l a , we obtain

ideal,

(a)r = e l ~ + e ~ + ( a 2 ) r This process

(a)r

(a- ela)s = a s - elas E (a)r+ el~ ~

is not a minimal

of the form for

~

c

n, we have

rings.

ring

104 iii) = iv). we have obtain ~(y

Let

O ~ , where each ~ ¢6A ~ , where ~ is [he semigroup

= C (~)

~ ~

~ ~

c ~ n ~ -- ~

~ =

Q ~

= 0

= 0

and

~

[J ~ a6A o~

is a regular semigroup,

each element of ~

= C(~ )

some

~ .

Consequently Let

Since for each

it follows that

nilpotent. and

~

~ = C(~), where 3.

Consequently

coincide w h i c h

Then

~

~

~

~

Hence for

there exists

Let

bi 6 % i

~

[RC(~)

Hence

~ (y ~-- I

so that

I

for some cannot be

~

be a semiprime ~

implies

~ha~

atomic ring and and

~

~

~

is atomic,

be its semigroup

is a regular ring.

~.

~.

By 6.1, we have

Hence by 5.9, we conclude that

such that

[~

is

~ =

~ ~ , where each ~ is simple and c~A c~ and we know that the ~ annihilate each ai ~ ~ i

a i = aibia i

and

~i

~ ~j

if

i ~ j,

and thus

= alblal + a 2 b 2 a 2 + ' ' ' + a n b n a n ... + a n ) ( b l + b 2 +

... + b n ) ( a l + a 2

+ ... + a n )

is a regular ring.

II.6.3 COROLLARY.

PROOF.

[R~. Let

By 1.13, the minimal right ideals of

together with the hypothesis

= (al+a2+

then

,

is s sum of elements of

I ~ ~ # 0.

a = a l + a 2 + ... + a n, where

al+a2+'''+an

and hence

~ ~ A,

~

Further,

c~

is semiprime.

atomic hence regular by 2.8 and 1.3.6,

=

, and for each

is completely 0-simple,

is a dense extension of

s dense extension of

a

[~.

is a primitive regular ideal of

has a nonzero intersection with

other.

is an ideal of

is an ideal of

~

Thus

We have seen in the proof of "iv) ~ i)" in 6.1 that every nonzero ideal

PROOF.

of

~

~

~ . c~

A part of the proof of "v) = iv)" in 2.8 can be

~ ~ ~ and :~A c~

II.6.2 COROLLARY. socle.

~ # ~, we

Hence in particular,

~ E A, ~ ~

that each element of

applied to the present situation showing that ~ =

For

By 2.8,

N = O(~).

be a nonzero ideal of

~ A, where

~

is actually an orthogonal sum of

is a sum of elements of some

by 2.8; it follows

iv) = i). I

~

socle of

since the sum is direct.

~ =

is a primitive regular semigroup. and

is a simple atomic ring.

If

~

is a primitive regular ideal of

is a dense extension of

~L~

for a ring

3,

~.

This follows immediately from 6.1 and 6.2.

The next corollary is known as the First W e d d e r b u r n

(or W e d d e r b u r n - A r t i n )

Theorem w i t h "semisimple" instead of "semiprime" w h i c h are in this case equivalent. For other c h a r a c t e r i z a t i o n s of these rings, Jans ([i], Chapter 2). II.6.4 COROLLARY.

Also consult Behrens A ring

~

see Fuchs and Szele

[i], G o l d m a n

[i],

([i], Chapter III, Section 3).

is semiprime and satisfies d.c.c,

for right ideals

if and only if it is a direct sum of a finite number of matrix rings over d i v i s i o n rings.

105 PROOF.

~ 6 , where each ~ is simple c~A~ atomic. Since the sum is direct, every right ideal of ~ is also a right ideal (y of 3. Hence ~ satisfies d.c.c, for right ideals, which by 2.8 and 1.3.14

implies

Necessity.

that

infinite,

By 6.1, we have

% ~ ~ffn ' n

X n

~ =

matrices

over the division

then for any infinite sequence

obtain an infinite d e s c e n d i n g chain Therefore

A

ring

~cl.

~i,c~2,... ~ A, letting

Jl ~ J2 ~

If

A

is

Ji =

~ ~ we k>i ~k' "''' c o n t r a d i c t i n g the hypothesis.

must be finite

The proof of the s u f f i c i e n c y is left as an exercise. II.6.5 EXERCISES. i) for

Prove that each of the following conditions on a ring

~

~

is sufficient

to be a division ring. a)

[F~

b)

~

is a primitive regular semigroup.

c)

For every

is a regular ring with only one nonzero idempotent. 0 # e ~ ~,

there exists a unique

x ~ ~

such that

= axa.

(These conditions are trivially necessary.) ii)

Let

ideal of

~ =

~ ~ , w h e r e each ~ ~A ~ is a direct sum of some ~

~ a)

A

b)

each

is a simple atomic ring. and conversely.

is finite if and only if ~

is isomorphic

only if every minimal

~

Show that every

Deduce that

satisfies d.c,c,

for two-sided ideals,

to a matrix ring over a division ring if and

two-sided ideal of

II.7

~

satisfies d.c.c,

for right ideals.

ISOMORPHISMS

We consider first isomorphisms of some of the rings we have encountered in the two preceding sections.

After that we give a construction of all isomorphisms of

two Rees m a t r i x rings. 11.7.1 LEMMA. >emigroup socles of

Let

~

and

~

and

~

w h i c h is a division ring.

~

and = of

~

~

G

~ ~

~

and

~

of

~/ =

~

~

onto

onto

AI

and

~

onto

~ ~/t

onto

~.

= ~I~ ' we have an i s o m o r p h i s m

~

~ ~ = ~~ of

~i

and

~

b e the

has no ideal can be uniquely

By 6.1, we have

is a simple atomic ring.

~ and ~ ~ ~ It is easy to see that

such that

~

~.

@ ~ /, w h e r e

~IEA~ respectively. A

~

be an isomorphism of

~//, where each

aEA ~ % and ~11,

function ~

Let

and assume that

Then every isomorphism o f

extended to a ring isomorphism o f PROOF.

b~e semiprime atomic rings, ~

respectively,

~

onto

~ aEA ~

Hence

are the semigronp socles ~

for all ~i~ .

~ =

induces a one-to-one ~ E A.

Letting

106 i ,V )I , where ~--~U (h(~,V) and r~~ ~--:---~U,(£<~ dim V(~> i c~ (y in view of the hypothesis on ~. We know from 1.5.12 that every isomorphism @

By 2.8, we have

$2~U(£,V)

onto

isomorphism

@

restriction

of

SU(&,V)

onto

~'

.

~2,UJ(h',V'), of

~U(h,V)

~

to

where

onto

SU(L,V)

that

Since

'~a

admits

~ =

~ ~

r = r I + . . . + r n,

hence

of

~

extended

Using 1.5.12 again,

is the unique extension

a uniqu~ ~xtension

the function +

morphism

l, can be uniquely

•'u'(A''V')"

..

+

,~:r ~ rl~a I where

dim V >

of

@

of

to an

we see that the

to an isomorphism

of

SU j(A',v'),

We deduce onto

~

~

onto

~

to an isomorphism

defined

~'

Let

rn~ n

At each step,

~

c~

-

.

extension

of

by

r.i % '~'-i' is a unique extension

is the unique

Ii.7.2 LEMMA.

~

of

~

the extension

of each

isomorphigms

to an isomorphism

be an isomorphism

~c

of a rin$

of

[~

~

to an isoare unique,

onto

~'

onto a rin~

~'

Then

the function

e-:(~,o) ~x = [ k ( x e - l ) ] G ,

~vhere

onto

~(.~) PROOF.

-

(k,~)

((>..,0)

x0 = [(x6-l)p]G

w ~ = .~, r r6 ()-,0) ( ~ ( ~ ) and

(x ~ ~ ' ) ~

such t h a t

(r t ~).

For

x,y Q ~',

[(xy)

= {.),l(xy)e-£]}e

= [~[(x+y)6-1]]6 =

so that

~

x(ly) = x[k(ye-t)]6 =

which proves

that (~$)x

and similarly one-to-one From

=

},r x

=

and

x ( ~',

Ikr(X@-l)}@

= kx+Xy,

p-

is a right

translation;

further

= [ (x6-l) [;, ( y e -i ) ] ; e

= [(×e-1)p]ey

If also

that

= (x-6)y ,

(~2,~) ~ ~(~),

= {),[
x(~,7) = x%0-~, showing easily

= (fx)y,

= [>(x6-1+y6-1)]e

analogously

lL[~(xe-1)]e]

and onto follows r ( ~

= lX(xe-1)]ey

{.[(xe-i)oj(ye-1)]e

(~,p) ~ fZ(~').

o__[f N(~)

we o b t a i n

[k(xe-1)J6+[),(ye-i)]e

is a left translation,

is an i s o m o r p h i s m

= {k[(xe-l)(y6-£)]]6

= [[x(,,,:e-l)](ye-l).~e ~(x+y)

~ ~(~)),

~

then

= X-6x

is a homomorphism.

from the fact

that

@

we obtain

= [r(xs-l)j6 = (r@)x = krsX

shares

That

@

is also

these properties.

i07

so that

--Xr= Xr~, and analogously

II.7.3 LEMMA.

Let

~

Pr = PrG --

which proves

be an essential extension of a ring

be a maximal essential extension of a ring onto

It

more~ X

I ~.

~

onto

~

if and onl~ if

~

~r ~ =

X

~r~"

I, ~(I) = 0, 3 ~

Then ever~ isomorphism

ca_nn.be uniquely . . extended . to an isomorphism maps

the formula

__°f ~

into

~

of

~.

1

Further-

is a maximal essential extension of

I. PROOF.

Let

8

be an isomorphism of

the ring analogue of 1.7.5 implies that ~(I), that all

and ~

T' = 7(3t:I l)

Hence

onto

X = T~Tt-I

~(I)

It

m = T(3:I)

is an isomorphism of

is an isomorphism of

x E I.

I

onto

~(I ~) = 0

and hence

is an isomorphism of

3'

~(I')

Then

onto

~(I').

with the property

is an isomorphism of

~

into

3

into

Now 7.2 asserts

~t

~x ~ = nx8

for

such that for all

x ~ I, we have x~ so that

X

extends

Let also for any

~

r ~ 3

x0

~x@ T ~.

be an extension of and

6

to an isomorphism of

~

into

3 ~.

Then

x ~ I, we obtain

(r~)(x@) = (r~)(xw) = (rx)~ = (rx)x = (rx)(x~) = (rx)(x~) and similarly

(xS)(r~) = (x~)(rA).

r ~ - rk { ~ t ( I t ) .

Bun

~,(I')

~,Cl

t) e ~' = ~C~'} = o

~ t ( I t) = 0.

x ~ I

is an ideal of

which together with the hypothesis implies

Since

that

Consequently

3'

is arbitrary,

~'

it follows that

satisfying

is an essential extension of

r~ = rx

and thus

~ = X

since

It

r ~ ~

is

arbitrary. The necessity in the last statement of the lemma is obvious; from the fact that

~X

is an essential extension if

sufficiency follows

I'.

The next result is new. II.7.4 THEOREM. socle

~

Let

~

be a semiprime tins essential extension of its nonzero

and assume that

3

has no ideal which is a division ring.

ring which morphism

is a maximal essential extension of its socle ~

of the semigroup socle

~

of

~

onto

can b e uniquely extended to a rin$ i s o m o r p h i s m X

~

~

onto

PROOF.

Let

3 ~ ~

if and only if

Since

According ~.

~t = C ( ~ ) ,

to 7.1, ~

By 7.3, ~

R

Hence

the semigroup socle o__[f 3

onto Rt

3 t

admits a unique extension

Rt

be a

into

3 ~.

~

of

~t

Furthermore, ~.

It follows from 5.9 that

is a primitive regular ideal of

by 6.1 we conclude that

can be uniquely extended

Let

Then every iso-

is a maximal essential extension of

be an isomorphism of

is a primitive regular semigroup. ~t

3

X

~.

~t

is a semiprime atomic ring.

to a ring isomorphism

to an isomorphism

X

of

3

0

of

into

~ ~t

onto

108

Consequently If

~

~=

X

provides an extension of

is another such extension of ~'.

But then

=

~I~

The last statement

8

~ = C(R)

• = X

and

if every m u l t i p l i c a t i v e

~/ = ~(~')

exercise

i), a ring

~

A semiprime rin$

~

~

~F~

onto

~I~ j

for some ring

~

that

~'

nonzero socle.

By the ring analogue of 1.7.5, we may suppose that

in a maximal essential

extension

the proof of uniqueness

~

of the socle ~,

~"

6'

satisfies

in 7.3, it follows

X

of

~ = X

of

~

such that

£

and

BA

~'

Hence

is embedded ~ = ~I~'

~J~.

Similarly as in

(x~"~ R, r E ~).

~', we have that

which proves that

is said to be invertible

AB

3'

into

R' ~

T r~ = T rX, where

by 5.9 which by 1.7.5 yields is additive.

We now consider isomorphisms of Rees matrix rings. division ring

ac-

that

is a dense extension of

Consequently

Since

the conditions of 7.4 and ~ e n c e

(xg)(r~) = (xg)(rk)

be the semigroup socle of But

[4].

is a semiprime ring essential extension of its

to an isomorphism

(r~p)(xg) = (rk)(x¢),

~'.

has can be characterized m u l t i p l i c a t i v e l y

is the semigroup socle of

admits a unique extension

£

An I X I'-matrix

B

if there exists an I' X l-matrix

over a A

over

are the identity matrices of the c o r r e s p o n d i n g sizes.

The following p r o p o s i t i o n is due to Hotzel

[l], its proof appears here for the first

time. 11.7.6 PROPOSITION. matrix

rings.

X = (xij)

Let

over

h

~

Let

~ = ~(I,h,A;P)

be an i s o m o r p h i s m o__ff g

we write

finite A × A ~ - m a t r i x over &', an___ddsuppose that

X~ = (xij~).

h', B

P$ = AP~B.

i s an i s o m o r p h i s m o f

~

onto

and

~' =

onto

~(I',hJ,A';P ')

Next let

A

be an invertible row

~J,

Then the function

X

defined by

(X E ~) Conversely ever~ isomorphism of

can be so constructed. To prove the direct part, we let (X*Y)X

= B(X*Y)$A

be Rees

g', and fo____rran____yymatrix

be an invertible column finite I'X I-matrix over

x:X ~ B(X$)A

PROOF.

X"

extension of its nonzero socle, has unique addition.

be an isomorphism of that

r~0 = r k.

and

which has no ideal which is s division

it follows

Rt

e

(snd is thus a

cording to 5.9,

T = T(~":R').

imply that

[i], see also Gluskin

a ring with the properties

Letting

~¢.

has unique addition if and only

isomorphism onto another ring is additive

rin$, a n d is an essential

~

into

by the uniqueness of both

The next corollsry is due to Rickart

II.7.5 COROLLARY.

where

~

follows immedistely from 7.3.

ring isomorphism).

Let

to an isomorphism of

~, then

and hence

Recall that by 1.5.22,

PROOF.

9

X,Y 6 •

and obtain

= B(XPY)$A

= B(X~)(P$)(Y$)A = B(X$)(AP'B)(Y$)A

(XX)P'(Y X) :

(X~) * (Yx) ,

~

onto

~

109 Since B

X

is obviously

easily

implies

additive,

that

For the converse, division assume

ring,

that

k

maps

we let

the statement

~

it is a homomorphism. ~

k

onto

~'

be an isomorphism

in the proposition

is not a division

The imvertibility

of

A

and

and is one-to-one. of

~

onto

is trivially

ring and consider

~'.

If

satisfied.

the commutative

~

is a

We thus

diagram

X

au(a,v) . where

the vertical

2.8, and of

{(~,a)

(U,h,V)

the isomorphisms

is the isomorphism

onto

of describing exploiting

arrows denote

(U',h',V')

in the proof of "iv) = vi)" in

induced by the semilinear

provided by 1.5.13.

the isomorphism

the commutativity

, au, (k' ,v')

X

isomorphism

(~,a)

The idea of the proof consists

as in the statement

of the proposition

by

of the above diagram.

Let

z=iz~IxeA], z' = { z ' t be bases of

V, U, V'

I X ' E Ar}

I

and

w'={w~,li'E

U ~, r e s p e c t i v e l y .

(m,a)

linear isomorphism

w=iw iliE f], Let

I' } b

be t h e a d j o i n t

of

the

semi-

and l e t

A = (ceXX,)kEA,X,EA,,

B = (~i,i)i,Ei,,i~i

where =

~ z' k'6h '~kk' ~''

z a Then

A

is row finite,

size and are invertible

B

is column

since both

b-lw i =

finite and both a

and

b -I

(1)

~ w' i'£f' i'll'i" A

and

B

are one-to-one

have the required and onto.

Further,

we obtain Pki w = (Zk,Wi)tO = (zk,b(b-lwi))~

= ( ~

~,z'

X'EA'

r

= (z a,b-lwi )I

w'

i'' i'~l'

'

i'~i~i)

Wj I k'' i ') ~i'l

Z /

=

= which

~ X'EA'

~ ~k'( i'El I

E

in terms of matrices We also have

E

zi,s -I =

(2)

'Px

]~'6A' i'{l '(~xk

''i'~i'i

has the form ~ %E A °k 'Xz k

P~ = AP'B

as required.

so that by (1)

we obtain

110

z!

z' a -I ~EA

= ~-~ ( ° k ' x '~) lEA which

XEA

£~ , ~ k ~ ' z "

~

~' i

•

~

(~

~'EA'

,)z',

(o, , . ~ ) ~

)EA

~ ~

~

implies

where

(o%,X~)~k , = 61, ~, (t',~' E i'), (3) X£A i s the Kronecker d e l t a f u n c t i o n . I t follows from (2) and (3) that

6k~p/

¢

XEA ~'6A' i

XEA

I' !

~'EA'

i'61' ~EA

A

(4)

= i'EI' ~J P'' i i ~ i 'i" Finally

let

X = (xix) E ~{. X

~

PX

=

Following

( Y], p . . x .

).

iEl ~i ~ and hence we must express

(z{,a-l)ea

a

-i

ca

the above diagram,

A,D6 i

in matrix

~

e

we have

a -I ~

form.

(5)

ca Using

(i) and (4), we obtain

= (~ol,k(zkc))~ lEA = ( ~
= ~

~ (a.,.p..×.

) ~ ( z a)

= iEl ~ ( )~6A ~ (<~k,kPki)tv) (xiotu) (zPa)

which

=

~ ( i ~t iEl

=

L X'6A'

in matrix notation becomes

isomorphisms

E i'EI'

Z

C~ . , z J , ) A

).'EA' PA

<" (px,i,~i,i ' , ' z. (x ipw)C~pl,)z l, iEl

P'B(Xco)A.

Hence

continuing

the sequence

of

in (5), we have

a proving

~PiJ ,)i / ~(i , i )E( x i1

that

-i

ca ~ P'B(Xo%)A - B(X&)A

X X = B(X~)A.

II.7.7 COROLLARY.

If

g ~ g', card i = card A'.

~(I,h,A;P) Further,

e~ ~(I',A',A';P'),

hl ~-~ h{,

then

if and only if

card I = card I', h ~ g~,

card I

= card I'. II.7.8

i)

EXERCISES,

Show

that an essential

extension

of a semiprime

that every semiprime

ring can be embedded

the same for "prime"

instead of "semiprime".

ring is semiprime.

into a semiprime

ring with

Deduce

identity.

Do

iii ii)

Let

A

has no nonzero

iii) each

Let

~

be a m i n i m a l nilpotent

~c~A

ideal then

t~ ,}0/~ A,

and

is a simple

right

elements,

atomic

ring.

of a semiprime A

must

be two

ring

~.

be a d i v i s i o n

families

of rings

For an i s o m o r p h i s m

each (r)

that

there exists

~ ~ A, an i s o m o r p h i s m E

II ~ c~A a)

, we have substitute

b)

assume

c)

combine

([5], Chapter

of

a complete

items

L~

~

function

onto

~ = r ~ .

that each

For a d i s c u s s i o n petrich

~

(r),~

a one-to-one

Also

direct

~

of

~ ~ c~6.A o~

~

of

A

such

consider

that

is a m a x i m a l

primitive

if

A of

that

onto

onto

A~

and for

for every

the following

sum by a direct

that

and assume

~

c~ II ~ j, show ~ 6 A ~ o~

Show

ring and an ideal

variants:

sum, ring,

a) and b).

of isomorphisms V, S e c t i o n

2),

of

and

the translational for a survey

hull of a semigroup,

consult

petrich

[3].

see

112

PART III

L I N E A R L Y T O P O L O G I Z E D V E C T O R SPACES AND RINGS This is essentially a topological

treatment of some of the rings studied

previously provided with a topology induced either by the vector space or by multiplication.

For a pair of dual vector spaces

U-topology on

V

(U,V)

over a division ring

is discussed here in considerable detail.

Several properties of

subspaces and of semilinear

transformations

It is proved that

the finite topology is a completion of

U-topology.

U*

with

Linearly

relative

topologies

to this topology are e s t a b l i s h e ~

compact modules are shown to be complete,

vector spaces are c h a r a c t e r i z e d in several ways. for the ring of all continuous

the ring of all continuous

4, the

V

with the

and linearly compact

The finite and the uniform

linear transformations are discussed.

For

linear transformations of finite rank various properties

of one-sided ideals relative

to the relativized finite topology are established.

A

completion of a semiprime ring which is an essential extension of its nonzero socle, with a topology and thus a uniformity III.l

induced by its socle,

A TOPOLOGY FOR A V E C T O R SPACE

For a pair of dual vector spaces induced in a natural way by

U

(U,V)

we will introduce a topology on

and the bilinear

general topology will be assumed; follow Kelley

is constructed.

form.

V

A rudimentary knowledge of

for the terminology and basic results we will

[i].

The C a r t e s i a n product of a family of sets the particular case when

for all

is denoted by XA

In

instead of

X. We will denote a topological

T

~ E A, we write

~ X . o6A ~

~ X , ~A ~ since in this case we are actually dealing w i t h the set of all functions from A into

X = X

~X } ~ A ~

is a topology on

simply write

X

X

space by

(X,T)

where

X

is a nonempty set and

given as a collection of (open) sets;

sometimes we will

if there is no danger of confusion.

For a nonempty family

IX } ~ A

of topological spaces,

their ~roduct is the

topological space on the set the sets

U = ~ (x)

fixed

and

~

U~

Ix

~ X with the topology whose subbase consists of c~A ~ E U~} where ~ E A and Up is an open set in X~ for

(it suffices

to take

U~

ranging over either a base or a subbase

113 of

X~).

This topology

topological

is called

the product

(or Tychonoff)

We now take each has a subbase

~ X . o~A ~ to have the discrete

topology and the

space is denoted by X

consisting

topology.

Then the product

~ X aEA ~

of the sets

S(~;y) = i(x ) I x~ = y], where

~ E A

product E A

and

y E X~.

If

X~ = X

for all

topology by taking all sets of the form and

y E X.

of all functions

Specializing

from

X

~ E A, we obtain s subbase

for the

S(~;y) = if E X A I ~f = Y}

where

further by letting

into itself,

A = X, we obtain

and a subbase

X X,

the set

for it is given by the sets

S(x;y ) = [f E X X I xf = y}. The following

concepts

Ill.l.l DEFINITION. X A, where

A

are due to Jscobson

For

A

has the discrete

Hence s base

and

X

nonempty

topology,

for the finite

[5]. sets,

is the finite

topology

consists

the product

topology of

topology on X A,

of the sets

= ~f ~ x A I ~i f = x i'

i = 1,2 ..... n}

B(~I, ...,~n;Xl,...,XnJ n = N B i=l ((~i;xi )" 111.1.2 DEFINITION. topology.

Let

Then the finite

topology of

(A,V)

topology of

be a vector space and give ~V

relstivized

spaces

(U,g,V)

to

V*

A

the discrete

is the finite

V*.

We now fix s pair of dual vector the image of

U

in

V*

under

the isomorphism

and recall

f:u ~ fu' where

that

nat U

Vfu = (v,u)

is (vE V),

see I.l.l. 111.1.3 DEFINITION. topology on U

into

U

V*, where

Equivalently, where

For

for which

the latter

as open sets in

U

V*).

symmetrically

the finite

If needed

of

U

is the

f

(or

be a homeomorphism finite

topology of

K = L N nat U

where

the U-topology

of

L V

of V*.

U

onto

nat U,

This amounts

Kf -I, where

K

of

to

ranges over

runs over all open sets in by considering

nat V

in

U*

topology.

111.1.4 NOTATION. V.

the V-topology

just the sets of the form

nat U

with

that

the relativized

all open sets in One defines

a dual pair,

f:u ~ f u (u E U) is a homeomorphism is endowed with the finite topology.

we may require

the latter is given

declaring

(U,&,V)

the isomorphism

For

(U,&,V)

for emphasis,

we write

a dual pair, TU(~,V )

Tu(V )

instead of

denotes ~u(V).

the U-topolosy o f

i14 111.1.5 LEMMA.

A base for the finite

topology

of

V*

consists

of all sets of

the form B(Vl,V2 ..... Vn;61,62 .... ,6n) = i f E V* I vif = ~i' where each set PROOF.

~Vl,V2,...,Vn}

From above,

linear independence. set

B(vl,v2,...

is linearly

i = 1,2 ..... n}

indel~endent.

a base is given by all such sets without

The proof

,Vn~.$i,~2 '

that for

..,6n}

B(zl,z2,...,Zk;~l,V2,...,~k )

iVl,V2,...,Vn}

is either empty or equal

with

~Zl,Z2,...,Zk]

the requirement

linearly dependent,

of

the

to some

linearly

independent,

is left as

an exercise. III.i.6 COROLLARY.

A base

for

Tu(V )

C

consists

of a!l sets

= iv ~ V I (v'u i) = ~i'

i = 1,2

(Ul,U2,...,Un;61,62,...,6n) where

{Ul,U2,...,Un} PROOF.

is a linearly

If W For

A

of

For a dual pair

= 0

for all

v ~ S},

T~=

Iv~

Vl(v,u)

~ 0

for all

u~

and

subset of a vector

A; we also write u E U, we write

on

binary operation -i x ~ x

making

W,

r}. then

[Wl,W2,...,Wn]

[A]

denotes

instead of

Tu(V )

U, di___m_mT < ~}

the subspace o f

[[Wl,W2,...,Wn}].

of

0

ha___~s

as an open base,

(x,y) - x • y

of

G.

(G,',T)

it a group,

is continuous.

consisting

and a topology

is jointly

continuous

In such a ease,

We write

(G,T)

T

in a topological

to give the neighborhood

G.

the preceding

G

G, a binary for which

and the operation

G

Since

the

of in-

if there is no need for

for any two elements

of

in order

system

on

is said to be compatible with

translation

group these are homeomorphisms,

it suffices of

a left and a right

of a set

T

or simply

the group operation or the topology.

x,y 6 G, there exists

reason,

T

v m = iv] z, u m = [u] m

group is a triple

the group structure

the identity

and

as an open subbase.

G

version

group,

V

Exercise.

A topological

emphasizing

space

The neighborhood ~

subspace o f

[u ~ I u ~ U]

operation

U.

! subspace o f

u l(v,u)

{ T~ I r

PROOF.

S

~u~

111.1.8 COROLLARY.

ii)

(U,V),

s ~=

generated by

i)

subset of

U, let

is a nonempty

v E V

independent

Exercise.

Iii. I.7 NOTATION. subspace

n} '''''

G

sending

x

onto

to give a topology on a

(or a base or subbase

We will use these facts without corollary will be particularly

y, and

express mention. useful.

thereof)

For this

of

i15 III.l.9 COROLLARY.

Th_~e neighborhood

system of

v E V

Tu(V)

for the topology

has i)

{ v + T z IT

ii)

~V+U~

subspace

lu E U}

III.l.10 LEMMA.

U, di___mmT < ~}

as an o~en base,

as an open subbase.

In a topological

group

(G,T)

the following

statements

are

equivalent. i)

T

ii)

is a Hausdorff

topology.

The intersection

o f a l l neighborhoods

The identity o f

G

of the identity of

G

equals

the

identity. iii)

has a neighborhood

base

(subbase)

whose intersection

is the identity. PROOF.

Exercise

(see the remarks

IIl.l.ll LEMMA. G

Every

is a homeomorphism o f PROOF.

If

H

xH

each of which

vector space

is a topological V

making

viz.

A

i) ii) iii)

of

H

in

G

is

where

(with the obvious

(6,V)

is a vector

definition),

T

is a

and such that the

is jointly continuous.

A subset

special

case, which will be assumed

topology. H

subspsce

The following

(L,V,T)

group under addition

only in the following

a maximal

III.i.14 LEMMA. V

ring

has the discrete

III.i.13 DEFINITION. is under inclusion

division

group is closed.

G, then the complement

is a triple

(6,v) ~ 6v

We are interested

space

group

is open by i.ii.

it a topological

scalar multiplication

throughout,

in a topological

Every open subgroup o f ~ topological

is an open subgroup of

A topological

topology on

translation

onto itself.

Exercise.

the union of cosets

space, &

left and every right

G

III.I.12 COROLLARY. PROOF.

above).

of a vector space of

V

is a hyperplane

if

H

V.

statements

concerning a subspace

S

of a vector

are equivalent. S

is a hyperplane.

dim(V/S) S = fm

= 1. for some

0 # f ~ V*, where

fZ

is taken relative

to the dual pair

(V*,V). PROOF. between

The equivalence

subspaces

of

V

of i) and ii) follows

containing

S

from the well-known

and subspaces

of

V/S.

correspondence

116 i) = iii). where

Let

v

be a vector in

V

the sum is direct in the group sense,

x E V, we can write easily that

f

x = x+

is linear,

iii) = i).

Let

v

(xf)v so

not contained in since

for unique

f E V*

and

be a vector in

V

S

S.

Then

V = S ~

is a hyperplane.

x E S

and

xf E A.

Iv],

Hence for any

It follows

f~ = S. such that

vf = o # O.

Then for every

x E V, we can write x = (x- (xf)o-lv) + (xf)o-lv ~ f.L + Thus

v ~ f±

generated by

implies

f~" +

f~

v.

Hence

fm

For

(U,V)

a dual pair and

and

111.1.15 LEMMA. of

V, we have

Iv] = V, where

Iv]

is the subspace of

S

a finite d i m e n s i o n a l

is a basis of

S

S ~ S zz

and

for any subspace

v @ S, then the set

and

(vi,u) = 0

which implies that

THEOREM.

IIl.l.16

S mz ~ S

(A,V,Tu(V))

fu

na__!t U = [ f ~ V* I f

~) c

If

Consequently

x,y E V

(x+y)+T

If is

such that

(v,u) # 0

and

u E S~

topolo$ical vector space,

(fu:V ~ (v,u)

(v ~ V)).

and finite dimensional subspace

±,

6 E ~, then

T

- ( x + T ~) = ~ x + T ±, proving that V

V

Tu(V )

and in

Moreover,

of

Tu(V )

6 ( x + T ±) = 6 x + T z, where

6

[I u ~ = 0 uEU

is com-

so

is its own neighborhood. (A,V,Tu(V))

Since the bilinear form associated with

it follows that

U, we have

(i.e., addition and inversion are

the scalar m u l t i p l i c a t i o n is continuous,

topological vector space. degenerate,

But then

u E U

the additive $rou~) structure of

patible with the additive group structure of continuous).

V.

and the equality holds.

i_~ss_ H a u s d o r f f

are continuous

of

i__~ continuous}.

For any

(x+T'~)+(y+T

i = 1,2,...,n. Hence

topology compatible with

which all functions

PROOF.

for

v ~ S ram.

S

~V,Vl,V 2 ..... Vn}

linearly independent which by 1.2.3 implies the existence of

is the weakest

subspace

S ~ Z = S.

[Vl,V 2 ..... Vn}

(v,u) # 0

V

is a hyperplane.

It is easy to see that

PROOF.

f~ +

[v].

which by i.i0 implies that

is a

(U,V)

is non-

~u(V)

is

Hausdorff. In order to prove that open for any

6 E a.

fu

is continuous,

First note that

and

x E u ~, we obtain

z+x

E ~fu I

and thus

(z,u) = ~ z+ut

8fu I = ~v E V I (v,u) = 6}.

and

~-- ~ful

(x,u) = 0

proving that

'

-i 6f u

it suffices to show that

so that 6fu I

For

(z+x,u)

is

z E 6fu I

= ~.

Hence

is open and hence that

fu

is continuous. Next let of

V

be a topology on

and in which all

T-open since

group.

•

fu

is

fu

V

compatible with

are continuous.

continuous.

But t h e n

For

the additive group structure

u E U, the set

7u(V ) ~ T

since

(V,~)

u z = Of u-I

is

is a topological

117 If

f E nat U, then

f = f u for some u E U and we have seen above that fu To prove the converse, we first let f be a nonzero element of V*

is continuous. for w h i c h v E V

fZ = Of -I

such that

subspace so that

T

is a closed subspace of

v ~ f~.

of

U

such that

T ± + fm # V.

T ~ ~ f~. implies

Since

V \ fZ

the dual pair

[f] = [f]m& c

(nat T) ~

that

f

f±

= nat T

(nat T) ~ ~

Consequently

implies that

cannot be continuous.

v ~ T ± + fz

is a hyperplane so we must have

by 1.15.

f ~ nat U

f # O, there exists

It follows easily that

(V*,V), we obtain

By contrapositive, we deduce that which implies

Since

there exists a finite dimensional

( v + T ±) N f~ = ~.

By 1.14 we know that

Considering

V.

is open,

This proves

[f]~

which

f 6 nat T c nat U.

0f -I

is not closed,

the last assertion of the

theorem.

We consider next a converse of the foregoing theorem. III.l.17 LEMMA.

l__nn~ topological vector space

ation of continuous PROOF.

It suffices

E h, both we let

6 E g

Since N ~

f +g

f

and

and

M

of

Consequently

Then

which are continuous and

Assuming that

vf+vg

and thus

such that

vf+Mg

= (xf+vg)+

N

Nf + v g = 6.

(vf+xg)-

f,g E V*

= 6, so that

there exists a n e i g h b o r h o o d Nf = 6 - vg

v

x(f+g)

f,g E V*

are continuous.

v E 6(f+g)-I

Hence

a neighborhood

f+g

to show that for fq

is continuous,

(6 - vg) f-l-

(A,V,T), every linear combin-

linear forms is continuous.

of

= $.

are continuous,

vf = ~ - v g v

such that

Similarly,

For any

E h.

there exists

x E N ~ M, we now have

6 = 26 - 6 = 6.

v E N n M ~ ~ ( f + g)-l, proving that

6(f+g)-i

is open and thus

is continuous. Next let

tinuous.

If

open since

q E h.

If

q # O, then for any f

is continuous,

III.i.18 THEOREM. ha___~san open subbase satisfying

~ = O, then

~

N H = 0

Let

fq

is the zero function and hence con-

6 E 4, we have

proving that (h,V,T)

fo

~(fo) -I = (6o-l)f -I

which is

is continuous.

be a Lopological vector space.

fo_~rthe n e i g h b o r h o o d > y s t e m of

0

Assume that

consistin$ o f hyperplanes

and let

HEY

u ={f~ Then

U

is a t-subspace o f

v*l f

is continuous]

V*, T = Tu(V )

n U = { f E V* I fZ ~ A H. -i=l l PROOF. V = f m @ Iv]

First let

for any vector

6f -I= [ x E Since

H

H 6 ~.

Vlxf

and

for some

H i E ~].

By 1.14, we have

v ~ fm

Then for any

= ~] = { y + o v l y E

is open so is its translate

H = f~

H+qv

for some

f ~ V*

and thus

6 E g, we obtain

H, (~v)f = 6] = for any

o E h.

U (H+~v). But the union of open

118

sets is again open w h i c h implies that In view of 1.17, U

so that

-i

is a subspace of

implies the existence of f E U

6f

vf ~ 0

H t ~

ix open. V*.

such that

which proves

Let

v ~ H.

that

U

Hence

f

is continuous.

0 # v E V.

The hypothesis

By the above, H = f~-

is a t-subspace of

for some

V*.

-i For any H

H E ]~, we also have

is Tu(V)-open.

topology

Since

~

But the sets

"ru(V )

topology If

um

so t h a t

f E U, then

for some

f ~ U

which shows that

is a subbase of the neighborhood system of

T, it follows that

• -open.

H = fz = Of

~ ~ Tu(V ) .

Conversely,

if

u E U, then

0

u ± = 0u -I

form a subbasc of the n e i g h b o r h o o d system of

"ru(V ) ~ "r.

f~ = 0f -I

Consequently

for the

0

is

in the

T = "ru(V ) .

is an open neighborhood of

0

and hence must

contain a basic open set ~ H. with H i ~ ~. Conversely, let f E V* and assume i= 1 l n fi ~ U as above. that fz ~ N H i for some H i 6 ]~. Then H i = f+i for some i=l n Hence f~ ~ ~i f.~ = Ill,f2,... fn ]m But then -- i=l i ' . f E

[f]~" = f~C__ Ill,f2,...,fn ] ~ -

= [fl,f2,... ,fn] c_ U

in view of 1.15. 111.1.19 LEMMA.

Let

S

be a proper subspace of a vector space

and extend

A

to a basis

a basis of

S

a basis of

V/S.

Conversely,

if

A

B

o_~f V.

is a basis of

is s_ system of r.epresentatives of the cosets of is a basis of PROOF. then

v =

S

Then S, C

C = ib+S

V.

is basis of

makin~ u_£

Let

A

I b ¢ B\A} V/S

C, then

and

b_!e is D

B = A U D

V. Let

A, B

>-~ <~ibi biEB

v+S

showing that

C

and

C

be as in the first part of the lemma.

If

v ¢ V,

so that

=

generates

~ (Tibi+s = ~ o i ( b i + S ) , bi~B b .~B i V/S.

Assume that

~

oi(bi+S)

= S.

Then

b .EB\A i

~.b.

E S

and hence

biEB\A i i A. C

Since

B

~ o.b hi E B\A i i

is linearly independent,

must be a linear c o m b i n a t i o n of elements of this is possible only if all

is linearly independent and thus a basis of Now let

we obtain di E D that

v+S

and B

A, B, C =

D

be as in the second part of the lemma.

~ o i ( d i + s ) = ~ <sidi+S d.EDI dieD

s E S.

generates

and

Hence V.

s =

o i = O.

~ ria i ai~A

Suppose that

Hence

V/S.

so that

and thus

v =

v =

~ ~idi+s dieD

For

for some

~ Old i + ~ Tia i died aiEA

~ o.d + ~_~ Tia i = 0. dieD t i siVA

Then

v ~ V,

proving

119

oi(d i + S )

=

~ o.d i + S diED i

dieD where

d. + S l

E C

and

independence

of

C

independence

of

A.

111.1.20 of

S

III.l.21

by:

T = Tu(V )

ii)

T

iii)

has

The

ii) =

an open

S

all

oi = 0

that all

independent of a vector

conditions

i) =

ii),

iii).

This

by linear

~i = 0

and hence space

V,

on a topological

In the former = V.

case,

Then

by linear

a basis

of

V.

the c o d i m e n s i o n

vector

space

of

~.

thus

v + B

open.

Let

system of 1.19

Hence

by

Let

~

system

of

system of

0

the c o d i m e n s i o n

The statement

= H

or

0

consisting

from

so that

[I H)), 1.19

v E H.

on

0.

T

Hence

T~ c C

of

from 1.19

of V If

B

B E ~

B ~ ~ Then B

and

generated

by

exists

H

v

codim

~

implying B 6 ~

B = n

C = A U [v,x2,...,Xn] A U ~x2,...,XnJ

which

H

which

that = i.

T ~.

at least one

for

B E ~,

to a basis

Therefore

be a basis

is a h y p e r p l a n e

since of

V.

and

that

H

is

of the neighbor-

then

it follows

D =CU

the subspace

v ~ B

B c H

shows

~ X l , X 2 , . . . , x n} H.

Since

of a finite number

that

suppose

and

contain

is a hyperplane.

T c

Thus

dim V/G < ~

in

i < i < n,

such

and

as an open subbase

can be extended that for

codim G < ~

is a hyperplane.

B E ~, we have

contained

having

of a Let

dim V/(G ~[H) = dim V / G + i < ~.

Then for some

is the i n t e r s e c t i o n

is r~-open

there

of

n = i.

dim ( G / ( G n H)) = d i m V / H

be the set of all hyperplanes and

that

since

where

that

C D iXl,X2,...,Xi_l,Xi+l,...,Xn}

each

= V

for

codim (G ~ H) = codim G < ~.

follows

consisting

of the intersection

is trivial

suppose

G+H

~ G/(G ~h H)

be the topology

follows

that

(V/(G N H ) ) / ( G / ( G

H E ~

that each

f; H. HEY be a basis of V

V.

are hyperplanes,

G+H

so that

(G+H)/H

that any basis

It also

0 ~ v E

Hi

is an open n e i g h b o r h o o d ~

from

it follows

G ~ H

It then

Let

hood

generated

o_~

(~ B = 0. BEB 1.18 and 1.14.

from

to show

Then

V/G ~

iii) = ii).

for which

is finite.

where

V/H =

we have

dim G/(G A H) = i.

V.

U

for the ne.ishborhood

follows

It suffices

be a hyperplane.

Further,

~

codimension

... fl H n

H

t-subspace

subbase

G = H1 ~ H2 N

of

is linearly

'

for w h i c h

of h y p e r p l a n e s

H

B

following

finite number

~.

yields

For a subspace

for some

of finite

PROOF.

of

Hence

which

~ H = 0. H@~ has an o p e n base ~ fo_~r th_~e n e i s h h o r h o o d

~

subspaces

member

i ~ j.

~ Tia i = 0 siVA

= $

codim S = dim V/S.

THEOREM.

of hyperplanes

G+H

if

~ Tia + $ ai~A i

are equivalent.

i)

let

~ d. + S j

Therefore

DEFINITION.

is defined

(~,V,T)

d. + S I

so that

= -

of

V

B = ir~iHi ,

of members

T = T ~.

Assume

of that

N B = 0, Let A B~ Then the subspace

containing

B

so that

120 H E ~.

But

v ~ H, contradicting

ii) = i).

the assumption.

Therefore

N H = O. HE~

This is a part of 1.18.

III.i.22 EXERCISES. i)

Show that

Tu(V )

is the discrete

topology

if and only if

V

is finite

dimensional. ii)

Show

tivized

to

iii)

that for every finite dimensional

S

that for any

fl,f2,...,fn,f

vf. = 0 for i = 1,2,...,n, then i combination of fl,f2,...,fn.

is to be found in Dieudonn~

mation on this subject, (I6], Chapter

Section

see Behrens IX, Section

~ V*

satisfying:

topology.

We will

given relative

introduced

tacitly

to which

topolosy,

PROOF.

f

is a linear

and statements

proved

([i], Chapter

2), Bourbaki

6),(I7], Chapter

PROPERTIES

assume

II, Section II, Section

in this

For more infor([i], § I),

3 and Chapter

IV,

Conversely,

that

S

vigj = ~ij

is dense

vf = ~ ~ 0.

let

S

Since

vg = 6 ~ 0

in

of

S ~, S ~ ±

V*, where

V*

S

V*

and let

is dense we have

of

that V*.

S

is

the

Further

be a basic neighborhood

There exists

S ~ B(v;~ ) ~ ~.

of

let

f E V*

0, where

of

Letting V*.

and

Vl,V2,...,v n

By 1.2.3 there exist gl,g2,...,g n E S such that n i ~ i,j ! n. Set g = j~__igj(vjf+6j); then for any I ! i < n, we

vig = j~_Ll(vigj)(vjf+ 6j) = v i f + 6 i g - f E B(vl,v2,...,Vn;61,62,...,6n ).

g E S

is endowed with

is a t-subspace

have

that

(U,V)

etc. are defined.

0 ~ v C V.

independent. for

to the

that a dual pair

as well as

showing

be a t-subspace

B(vl,v2,...,Vn;61,62,...,6n ) are linearly

S

section

in relation

is dense if and only if it is a t-subspace.

Assume

such that

A subspace

V

OF SUBSPACES

of subspaces

in the whole

the topology on

g E S riB(v;6), we obtain

proving

rela-

v ~ V, if

[5].

TOPOLOGICAL

III.2.1 PROPOSITION.

so that

for any

[i],[2],13 ] and Jacobson

We consider here some simple properties

f 6 V*

V, Tu(V )

18). 111.2

finite

of

vf = O, we must have that

The origin of most of the concepts

Jacobson

S

is discrete.

Show

section

subspace

Consequently

(f+B(vl,v2,...,Vn;61,62,...,6n))

is dense in

V*.

N S

121 The following lemma is due to Dieudonn~ 111.2.2 LEMMA. PROOF.

For

S

! subspace o f

We denote the closure of

with

V, S ~

S

= {v E V I ( v + T z) n S ~ ¢

[i].

by

is the closure of

S.

S.

By 1.19, we have

for every subspace

T

of

U

(i)

dim T < ~},

also note that S aa

= {v ~ V I if

Now let s E S. where

v E S

Then

(s,u) = 0

from (2) that

s E S, then

and suppose that for some

( v + u m) N S # ~

(x,u) = 0

for all

and thus

by (1), say

u E U, we have

s E

(v,u) = (s,u) = 0

(v + u a) N S.

C

v ~ S + T ±.

of

V.

U

(V*,V). f

Then

f E V*

xf = 0

111.2.3 COROLLARY. PROOF.

for all

s = v+x u.

It follows

with

nat U

T~ ~

T

f

on

C

U

f E fame

~

T aa

x E S

v E V\S.

for which

S +T a

f

Hence by

( v + T a) ~ S

and extend

by letting

Also denote by

fa, where

for all

of

be a basis of

and let

xf = 0

=

B U {v} for all

its linear extension

to

is taken relative to the dual pair

= T

since

dim T < m.

which by (2) shows that

Every finite dimensional subspace o f

V

But then v ~ S~± .

is closed.

Exercise.

111.2.4 LEMMA. PROOF.

For an___~ysubspace

T

of

U, we have

T ~a~

= T ±,

Exercise.

III.2.5 COROLLARY. some subspace PROOF. Conversely,

B

arbitrary.

and

Using 1.15, we obtain

~ U, vf # 0, and

T If

of S

if

S

is closed,

S = T~

Let

Xl,X2,...,x n E V ~ ff.x. E T a. i=l I I

S

of

V

is closed if and only i f

S = T~

for

then

(S£) ~ = S

for some subspace =

T ~az

=

T~

=

T

by 2.2 with of

S~

s subspsce of

U.

U, then by 2 . 4 , w e have

S

is closed by 2.2.

III.2.6 LEMMA. PROOF.

A subspace

U.

S ±~

and thus

Let

Define a function

x E B, vf # 0, and otherwise V.

(s,u) = 0 Hence

by the assumption on

(I) there exists a finite dimensional subspace which implies that

all of

(2)

v E S~ m .

To prove the converse, we identify

to a basis

(v,u) = 0}.

If

T

is an n - d i m e n s i o n a l

{tl,t2,...,tn] such that

be a basis of

(xi,t j) = 6ij.

If

subspace o f

In particular, we have n

U, then

codim T ~ = n.

T. By 1.2.3, there exist n i~=l~i(xi+T a) = T a, then

n

0 = ( ~ ~kXk,ti) = ~ ffb(x' ~ti) = ffi k=l k=l ~

122

ix[ + Tj.} n =l

so that fixed

v ~ V,

let

is s l i n e a r l y

~i =

(v~ ti)

n (v - ~ ~ x., i= I ~ i Lj) = and

thus

v-

£ ~.x. E T z. i=l I 1

the v e c t o r s

x i +T ±

independent

for

(v,

i <

of v e c t o r s

V/'f ~

v+T

and

in

V / T m.

For

a

Then

n - 71 (v t.)(x. i= l ' 1 l'tj ) =

tj)

Consequently

generate

set

i < n.

(v,

tj)-

(v,tj)

0

=

~ =

~ @i(xi+T~) w h i c h p r o v e s that i=l form a basis. Hence codim T ±

thus

= dim V / T ~ = n. III.2.7

PROPOSITION.

The

followin$

conditions

o__nn_a s u b s p s c e

S

of

V

ar___~e

equivalent. i)

S

i__~sopen.

ii)

S

is c l o s e d

iii)

S = Tm

PROOF. Since

S

for

i) =

codim

some

finite

ii).

is open,

d i m T < oo

and

in any

it c o n t a i n s

Hence

T z c_ S

S < ~. dimensions[

topological a basic

which

subspace group,

open

implies

T

every

o__[f U. open

neighborhood

T = T m z - -~

Sm

of by

subgroup 0,

say

1.15,

is closed. T ~-, w h e r e

so

d i m S ~ < co.

By 2.2 and 2.6, we h a v e codim

ii) =

iii).

It

The b e g i n n i n g

of

V / T "L

which

is a l s o iii) =

i).

III.2.8

the

S = d i m V/S = dim V/S mm = d i m S z < ~.

follows proof by

from

2.5

of 2.6

that

shows

contrapositive

In s u c h

a case,

S

S = Tm

that

if

implies

The

basic

open

COROLLARY.

The

followin$

some

subspsce

is i n f i n i t e

that

is a b a s i c

COROLLARY.

for

T

T

open

is f i n i t e

neighborhood

neishborhoods

o__[f 0

T

of

dimensional, dimensional. of

are

U. then

0.

the o n l y

open

sub-

spaces. 111.2.9

conditions

on a s u b s p a c e

S

of

V

are

equivalent. i)

S

is a c l o s e d

ii)

S

is an o p e n

iii)

Also subspace

S = um

note

PROOF.

then

that

containing

III.2.10

a closed

for

hyperplane. hyperplsne.

some

a hyperplane S

and

COROLLARY.

f = fu E

nat U

so

S

is e i t h e r

thus m u s t

Every

Necessity.

hyperplane

0 # u E U.

be e i t h e r

subspace

In p a r t i c u l a r ,

of

for any

that by 2.9 we h a v e and

thus

V

nat U = V*.

that

closed S

or d e n s e

or

is c l o s e d

if and o n l y

0 ~ f E V*, F = u~

since

S z±

is a

V. if

nat

U = V*.

F = [v C V I vf = Oj

for some

0 # u ~ U.

But

is

123 Sufficiency. basis if

B

of

x E B\{v].

and obtain

Let

V.

f

Then

S c uz

hyperplanes

S

Let

u~

be a subspace of

f ~ V* = nat U

and

v ~ u m,

S

Let

S

uz

with

T

of

v

to a

and

u E U, we have

xf = 0 f = f

u

v~ >+s v { u~+S

for some

u~+S

V

and let

v E V \ S.

( v + T z) N S = @

be a basis of

=

is the intersection of (closed

0 # u E U.

U, we have

n

and thus

vf = i

is the intersection of some closed

be a closed subspace of

[Ul,U2,...,Un}

it follows that

Extend

for which

Every closed subspace o_~f V

finite dimensional subspace Let

v ~ V\S.

V

and thus must be closed.

III.2.11 LEMMA.

v ~ Tm+S.

and

and hence for some

Hence

and open) hyperplanes of the form PROOF.

V

be the linear form on

T.

Then for some

so that

Then

n

~ u++s =

=~ (u~+s)

i=l l

i 1

i,

u~-+S ~ V

Hence

and since

: u~,. i.e., S ~_ u~'i Consequently

u.a

is a hyperplane,

v ~ u+1 while

S c_ u~, as

required. The following p r o p o s i t i o n is due to Dieudonn6 111.2.12 PROPOSITION. topology of

V/S

If

S

[i].

is s closed subspace o f

coincides with

• .(V/S), where

V, then the quotient

(S~,V/S)

is a dual pair with

S the bilinear form PROOF. u ~ S~

If

(v+S,u) v+S

so that

(v+S,u)

If

v+S

(v,u) ~ 0

and thus

(v,u) # 0

(v,u),

} S, then (v + S , u )

v-v'

t S

and thus

(v-v',u)

= 0

for all

whicll shows that the function which to each pair

the value

so that

(v ~ V, u t Sin).

then

(v,u) = (v',u)

associates

bilinear.

that

= (v,u)

= v' + S ,

v ~ S ~ O.

is single valued.

For every

(v + S , u )

This function is clearly

so by 2.11 there exists

} 0.

u ~ S~

u ~ S ~, there exists

Consequently

such that v ( V

such

the new bilinear form is non-

degenerate. Let

m:V ~ V/S

topology of

V/S.

be the canonical h o m o m o r p h i s m and let A subbase for

~

~

denote the quotient

is given by

g = { A ~ V/S IA~ -l= u< u~ u], and a subbase for the topology

a

~

6 = Ts~(V/S ) ,

is given by

{u ° luE s~j,

where

u° = { v + s Then for

l(v+s,u) = 0] = { v + s l ( v , u )

= 0j = { v + s l v ~

u E S "~, we have

u~°-l={v~

vlv+s~

u °] = i v ~

vlv<

u~] = u<

u-).

124

so

u

o

E $

which

for some hence

u ~ U.

proves

that

Since

0 E u ~, it follows

u E u ± ± ~ S m.

g c g.

S C [.

The next III.2.13 space o f

V.

PROOF. that v {

if

Using

v ~ A,

= i.

is due

A+B

Let

A +

implies

that

For every

u E Am

A

(V,Uo)

= (v,u)-

( A + [v]) ~.

i)

(U,V)

ii)

Let

Now

of

(U,V)

B

be a finite

V

dimensional

suffices

v @ A.

sub-

(v,~)

( A + Iv]) ~ ±

to show

Then

u

= (v,u)-

x 6

suppose

that

= 0 Then

= (x- (X,Uo)V,U) But

then

[v]

the opposite

pair

that

E A ± and we may o u - Uo(V,U ) E A m and also

(X,Uo)(V,U)

inclusion

Prove

if and only

be a dual

B, it clearly

for some

let

~ A+

be a dual pair.

V'

of

x - (X,Uo)V E A m ~ = A.

A + [v];

EXERCISES.

Let

and

We suppose

(V,Uo)(V,U)

= (x,u)-

so that

( A + Iv]) m ± c

subspace

# 0

u E A m , we have

III.2.14

proper

be a closed

is closed.

x E A+(X,Uo)V and thus

[2].

on the d i m e n s i o n

[v]

0 = (x,U-Uo(V,U)) for all

u~] : u ° E

to M a c k e y

induction

then

u - u (v,u) E o

0 ~ A, so

is closed.

(v,U-Uo(V,U)) so that

-i Then At1 : u j" -i S c A~ : u ~" and

A E ~.

~ = 6.

PROPOSITION. Then

(Am) ~ = A

(V,Uo)

Therefore

result

that

let

Consequently

A = {v~Iv~ so that

Conversely,

that

if

always

(U,V')

holds.

is a dual

pair

for no

nat U = V*.

and write

$ = Su(V).

Prove

the following

statements. a)

The m a p p i n g S - S~

is an i n c l u s i o n

(S

inverting

function m a p p i n g

the lattice of all subspaces S~±

is a subspace

b)

S ~

c)

~%,$(~T(V))

d)

~r,$(~u(S))

e)

~T(V)

For

the dual pair

of

for every

of

the

V) lattice of all subspaces

of

into

U. subspace

= ~ ( T ~)

S

for every

of

V.

subspace

T

of

U.

= ~ ±(V) for every subspace S of V. S = is E $ IT A ~ Na} for every finite d i m e n s i o n a l

subspace

T

of

U. iii) V*

if and only

if

V

(V*,V),

is finite

prove

that

dimensional.

Tm± = T

for every

subspace

T

of

125 iv)

Let

(U,V)

complete lattice

be a dual pair.

(a lattice

l.u.b, and a g.l.b, v) (U/S~,S)

Let

in

(U,V)

Show that closed subspaces of

~u(V)

relativized

S

be a subspace of

to

L

has a

V.

Show that

~U/sZ(S)

is weaker

S.

for this section include Bourbaki

([6], Chapter IX, Sections

Sections 5,6), K~the

form a

L).

be a dual pair and

The references

V

is complete if every nonempty subset of

can be made into a dual pair in such a way that

than the topology

Jacobson

L

([i], § I), D~eudonne " ~ [i] , [2],

7,8),(/7], Chapter II, Section 4 and Chapter IV,

([i], ~ i0), Mackey

•2], O r n s t e i n

/i], R i b e n b o i m

([i],

Chapter Ill, Section 2).

III.3

T O P O L O G I C A L PROPERTIES OF S E M I L I N E A R T R A N S F O R M A T I O N S

For dual pairs

(U,V)

and

(UI,V ~)

and a semilinear t r a n s f o r m a t i o n

we examine here the conditions under which also one-to-one or onto. Dieudonn~

III.3.1 THEOREM.

The next theorem is due to J a c o b s o n

PROOF.

Let

(U,&,V)

(V,Tu(V))

into

Suppose that

i

and

transformation o__ff (&,V)

function from

into

a:V ~ V j,

is continuous and if so, w h e n it is [5], see also

[i].

a semilinear

function

a

by:

a

and

Then

s

if and only if

is continuous,

o,~ E &

be dual pairs and

(~,VI).

(V~,~u~(V~))

v ~ ~ = ( v a , u ~ ) ~ -I

&, and for any

( U I , ~ , V I)

into

and for every

(v E V).

Then

~

(~,a)

be

is a continuous a

has an adjoint.

u~ E U~

define a

is a function from

V

x,y E V, we obtain

(ox+Ty)~ ~ = ((~x+Ty)a,u')~w

-l

= ((o~)(xa) + (T~)(ya),ul)IW -I = (OW~-I)[(Xa,Ul)I~ -I] + (TW~-I)[(ya,ul)I~ -I]

= o(x~') +T(y~') proving that a w a

~

E V*.

Further, u~

is a composition of the following functions:

w h i c h is continuous by hypothesis, -i

which is continuous since both

continuous

function

b

which is T u ~ ( V ~ ) - c o n t i n u o u s by 1.16, and

and

g~

are discrete.

linear form which by 1.16 implies that

Consequently

u ~ E nat U.

Now define a

by bu ~ = u

For any

fu ~ g

if

u ~ = fu

--

v ~ V, u I E U ~, letting

(v,bu')w showing that

b

= (v,u)~

bu ~ = u, we have

= (Vfu)~

is the adjoint of

(u E U).

a.

= (v2')~

u ~ = fu

--

= (va,u')'~-l~

so that

= (va,u')

u~

is

126

Conversely, a

at

0,

b

be

the a d j o i n t

since

by

1.8

the

family

neighborhood

system

of

0

in

V',

open

For

in

and

let

V.

so that

(bu') m

v.

Let

x E

v E

u '~

v+x

E

a -i

is a n e i g h b o r h o o d

(bu')a;

then

implies

u'¢a -1

III.3.2

that

showing

[ u ~ m I u~ %

show

of

0

A function if

Cf

Dieudonn6

III.3.3

= 0

and

Let

spaces

Va = V ~ i)

only

note

Consequently,

va = 0 ~

using

a

of

2.2,

A

into A.

On

= 0.

completes

bu' ~ U

the o t h e r But

of hand,

then

the proof.

a topological

The

next

result

space

B

is due

to

is

that

if

bU'

is d e n s e

is o n e - t o - o n e any

u ~

pairs

adjoint

in

and

(~,a)

b.

b__ee

Relative

statements

to

hold.

U

and o_o_o~.

U, we h a v e

for all u' ~

u' E U'

U' ~

,v~}

for w h i c h ~

we

= v~,

u = bu'

for

some for

for uI ~

i =

in

1.5.4

first

be a b a s i s

via

(bU')-

is d e n s e Then

is open,

[v l'~v~,.

Let vi 6 V

with

( U ' , T v , ( U ' ) ) , th_~e f o l l o w i n $

0a -I = 0 ~

Va = V'.

b

be dual

(g~,V I)

v %

(1)

(bU')-.

we o b t a i n

bU j

that

b

(U',A',V')

into

for all

~

u = bu'

and

Uj

asserts

let

SI

of

SI

let

some

= 0

U.

and

S = u' {

such

that

be

b

is "one-to-one.

a finite for

dimensional

each

In subspace

1 ~ i < n, c h o o s e

[vl,v2,...,Vn].

a

Then

S 'm

that

(v~,ul) ' = 0

1,2,...,n,

since (v~,u l ) = 0 ~

we o b t a i n Sm

the

is

is a n e i g h b o r h o o d

= 0.

(v + x a , u ' ) '

for

u~ma -I

we h a v e

in

(va,u') t = 0

is o n e - t o - o n e

u ~ b(S '~)

and

C

and

for

= 0

(bU~) - ~ = U ~ Suppose

to s h o w

V~

vector

space

and

if

that

(v,bu')

ii)

a subbase

that

of

is ~ontinuous].

(£,V)

if and only

if and First

v = 0a -l ~

of

(U,TV(U))

is o n e - t o - o n e

PROOF.

order

continuity

u~ma -I ; h e r e

(bu') ±

which

(U,V),

set

(U~h,V)

transformation

a

ii)

open

v+

(xa,u')'

thus

u'~a -l

pair

a topological

for e v e r y

THEOREM.

topological i)

from

v + (bu') ~ ~_

(va,u')' a dual

constitutes to s h o w i n g

thus

and h e n c e

For

to p r o v e

[i].

a semilinear the

f

is o p e n

that

= 0

v + (bu') m c

U']

reduces

and

£u(V) : [a E £(v) la

open

It s u f f i c e s

(x,bu')

that

COROLLARY.

a.

the p r o o f

u,~ a -l , we w i l l

v ~

of

b(S ~ )

is an o p e n

= S m.

subspace

(via,u')1 Here of

U.

S

: 0

~

(vi,bul)

is a finite Hence

b(S 'm)

= 0 ~

dimensional is open,

(vi,u)

: 0

subspace and

thus

of

V

for any

so u

i

that ~

U I,

127

b(u~ + S ~ )

= bu ~ + b ( s 'm)

neighborhood system of Suppose roles of

that

a

b

and

is also open. u I in

v~ 6

in (i), we duduce

that

(b-10) ~.

b(v ~z)

Let

v £

v~

S

of

V

with

This

entails

But

the

together

for all

then

s'

E v 'a

v ~ E (Sa) ± ±

U.

(b-lo) ~ ~ Va

which

b

Va

Va

s'

Hence

E v ~a

together with

Va

,s

)'

since

111.3.4 COROLLARY. onto onto

PROOF.

With

(V~,Tu~(VI))

implies

there exists a finite di-

v ~ S.

Then

(v,bu ~) = 0

w h i c h implies

bu ~ = bs'.

= 0. Sa

(2) shows that

V ~.

and the hypothesis

(bul) m

v ~ E (b-10) m

Va

implies Now (2)

Hence

b(u ~ -s~)=0

(v~,u 'in

s~) ~ = 0.

2.2 shows

is finite dimensional. Va = (Va) ma.

By 2.2,

that Therefore

(Va) ± ±

is

is closed.

then interchanging

is dense in

the

b(v') a.

= (v

that

U~

for all

S ~

for which

that

(v',u')'

which proves

Interchanging

so that

S m ~ b(vlm).

the roles of

Finally,

if

must be both dense and closed and thus

(V,Tu(V)) (U,&,V)

of

implies

is closed.

Consequently

Sa~

hypothesis

is one-to-one,

we duduce that then

the

form a base of the

(2)

(va,u~) ~ = 0

v 6 S.

and hence by 1.15, v' E Sa

the closure of If

existence

with

Va

(Va) m = b-10

the property

and assume that

(bu~) ±

that

u ' + S ~±

is open.

is an open subspace of

bu' E ( b u ' ) m a ~

which

h

(b-10) z.

is an open subspace of

uI E U~

and thus

(Va) ~ z =

Then

mensional subspace

the sets

is open; we will show that

b

Va c Let

Since

U~, it follows that

b

and

b

in part i),

is both one-to-one and open,

Va = V ~.

the n o t a t i o n of 3.1, a if and only if

a

a

is a h o m e o m o r p h i s m o f

is a semilinear isomorphism o f

(U~,A',V~). Exercise.

111.3.5 EXERCISES. i) space and

Let

a

V ~.

be a semilinear transformation of a vector space Construct

(V~*,V~). ii)

Let

formation of

Then (U,V) V

U ~-) topology. For more

into

b

the adjoint

b

of

relative

is called the conjugate of

and

(U~,V ~)

V ~, and let

Show that

a

be dual pairs, V

is continuous

Chapter IV, Section 7), R i b e n b o i m

let

(respectively

information on this subject,

transformations,

a

a

V

into a vector

to the dual pairs and is denoted by

a

(V*,V) s*.

be a semilinear trans-

V ~) have the U - ( r e s p e c t i v e l y

if and only if see Dieudonn~

a*(nat U ~)

C

nat U.

[i],[2], J a c o b s o n

[5],([7],

([i], Chapter III, Section 2); on continuous

see Rosenberg and Zelinsky

[i].

linear

128 III.4 Before aboarding notions

related

Let

X

OF A VECTOR SPACE

the main subject of this secLion,

to completeness

be a nonempty

Cartesian product

COMPLETION

X × X.

of uniform

spaces,

set; a (binary)

we briefly

see Kelley

relation on

The set of all binary

X

summarize

([i], Chapter

the 6).

is a subset of the

relations

on

X

forms a semigroup

under the operation hop Further, and

= {(x,z) E X X X

X-I

I (x,y) ~ h

and

(y,z) E 9

for some

y E X].

is defined by X-I = [(x,y) 6 X × X I (y,x) E X} so that ( -i)-i = ~-i o The identity relation or the dia$onal is defined by

(~ o p)-i = p

~(X) = i(x,x) Ix ~ X}, and is usually denoted by

~.

For

~ ~ X XX

and

A ~ X,

we let k[A] = [y E X I (x,y) E k and for

A = ix}, we simply write

A uniformity satisfying

~

for a set

the following

%[x]. X

if

k E ~,

then

~ ~ &;

(~)

if

% E ~,

then

p o p ~ X

if

~ E ~,

(6)

if

~,0 E ~,

(~)

if

X E ~

The pair

completely

%

then

and

family

~

if

~ E ~, then

(6 ~)

the intersection

~,

space.

that

family of binary relations

on

X

p 6 ~;

then

p 6 ~.

A subfamily

a member of

~.

Note

One easily verifies

satisfies ~-i

(~),

contains

of

the following

on a set

~

is a base

for

X

assertion.

is a base for some

(~) and

a member of

of any two members

(X,~),

~

that in view of (¢), a base of

of

the topology

g

~

~, contains

a member of

of the uniformity

~.

~, briefly

the

is the family

{T ~ X I for every One shows easily

for some

of binary relations

if and only if

topolosY of

(k o p)[A] = k[p[A]].

X N p ~ ~;

~. ~

space

that

E ~;

contains

(V')

uniform

-i

is a uniform space.

~

For a uniform

It follows

is a nonempty

X ~ p ~ X xX,

determines

A nonempty uniformity

then

(X,~)

if each member of

x E AJ,

requirements:

(~)

(?)

for some

g

x E T

there exists

X E ~

is indeed a topology,

such that

and thus

(X,g)

%Ix] c T] . is a topological

129 A function mapping a uniform space uniformly continuous

if for each

(X,~)

into a uniform space

(Y,~)

is

p E ~, we have

{ (x,y) E X × X I (f(x),f(y)) E p] E ~. Further,

in the case that

formly continuous, (Y,~)

then

f f

is one-to-one,

onto and both

f

and

f-i

is a uniform isomorphism and the spaces

are uni-

(X,~)

and

are uniformly equivalent. As in the case of topological spaces,

(X,~)

is a uniform space,

of

~

to

of

(x,~)

if

Y

is reflexive, p ~ m

and

where

S

to be denoted by

transitive,

p ~

n.

A net

eventually in

A

D

[S n I n ~ D}

directed by

m E D

(X,~)

[Sn I n ~ D}

converges

p ~ D

D

(S,~),

Sn ~ A

for all

Sn E A

to a point

n ~ D; it is

whenever x

n > m.

relative to

~

(X,~)

is complete if every Cauchy net in the space converges to a point in the

(Sm,S n) 6 k

A uniform subspace of a uniform space

(X*,~*)

(i.e.,

(X,~)

f

m,n ~

k.

Further,

is dense if it is dense in is a pair

(f,(X*,~*)), where

is a uniform isomorphism onto a dense

X*; a completion is H a u s d o r f f if

the uniform topology is Hausdorff).

whenever

(X,~)

A completion of

is a complete uniform space and

subspsce of

if it

is a Cauchy net if for each

(X,~)

~.

A net

x.

there exists

the uniform topology of

if

such that

A net is a pair

~. t ~

space.

such that

D, directs

and is also denoted by

if

such that

in a uniform space

k ~ D

~,

A

and

is a uniform subspace

m,n E D, there exists

is eventually in every 3 - n e i g h b o r h o o d of i net

(Y,~)

is a d i r e c t e d set.

X

is the r e l a t i v i z a t i o n

on a nonempty set

is ij!n a set

if there exists

in a topological space

D

Y, and

>,

and if for any

In such a case

is a function on a set

[S n I n ~ D].

is a nonempty subset of

~ = [~ n (Y x Y) Ik E %0

Y, or the relative uniformity for

A binary relation,

(S,>)

the set

(X*,~*)

is a Hausdorff uniform space

A H a u s d o r f f completion of a H a u s d o r f f

uniform space is unique up to a uniform isomorphism and can be referred

to as the

completion. If identity

(G,g) 0

of

is an abelian topological group, G, let [N

N = [(x,y) t G x G I x - y

IN

is a neighborhood of

is a base for a uniformity with

~

(verify~).

can be omitted)

~

It follows

on

G

t N].

of the

0j

and the uniform topology for

in addition,

N

Then the family

that the topology of any topological

is a uniform topology.

refers to this uniformity;

for every neighborhood

~

coincides group

(abelian

The concept of a complete topological group specializing to a complete module, vector

space or a ring, we have in mind the uniformity of their additive structure. 111.4.1LEMMA. topology.

If

V

is a vector space,

then

V*

is complete in the finite

130

PROOF. gV

Since

is complete

([i]~ p. 192),

~

is discrete,

being

the Cartesian

in order

is a closed subspace convergent a net in

net in V*

relativized

topology,

by Kelley

to

f

and

with

Yfn = yf

V*.

Since

Hence

to the convergence

m E D

whenever

of

n such that

Xfn = xf

n ~ m 3.

V*

= vf

to the point it follows

whenever

(ox+Ty)f n

whenever

Finally

be

is the

of

is discrete,

vf

V*

~fn In ~ DJ

~vf n I n ~ DJ ~

that

mI

such that

let

the convergence

Since

such

to show that

to show that every

the topology of

([i], p. 217),

convergence).

m2

it suffices

=

n > m. (ox+Ty)f

n >_ m2;

[here

there exists

= ( o x + T y ) f n = ~(Xfn) + T ( y f n )

But then

f

= o(xf) + T ( y f )

is linear so that

f { V*.

Hence

V*

is closed and

complete.

thus

III.4.2 THEOREM. isomorphism where

~

of

V

Let

into

is the finite

PROOF.

nat V. V

By definition,

be a dual Eair and

Then

is a dense subspace

The function ~u(V)

f

subspace

x -y E T~ ~ (fx-fy)t [ (x,y) I x - y

T

U*, and

on

V

of

U

and

finite

is complete

f

of

V

in the onto

a homeomorphism

topology of

U*.

For

x,y E V, we obtain for all

t E T

is a uniform neighborhood

[(fx'fy ) I (fx - fy)t = 0

U*

isomorphism

which makes

the relativized

f t = (x -y,t) = 0 x-y = 0 for all t ~ T.

E T ~]

of

is an algebraic

is the topology

nat V, the latter having

any finite dimensional

f:v ~ f be the natural v is the completion of (V,Tu(V)) ,

(f,(U*,~))

topolosy.

By 2.1, nat V

onto

The set

(U,V)

U*.

finite topology by 4.1.

of

in

In view of Kelley

it suffices

([l] , p. 66),

f ~ 6 V,

spaces.

([i], p. 194),

the property

(ox+Ty)f n > m.

of complete

x,y C V, there exists

there exists

such that

m > ml,m2,m 3

whenever

(pointwise

v E V, there exists

n >_ ml; m3

is equivalent

v E V

Hence by Kelley

is complete,

to some point

product

o,T E &

exists

converges

to some

that for every For any

By Kelley

converges

for every

whenever

&V.

V*

which

{ fn I n h D} vf

product

to show that

of

V*

it is complete.

for all

in

V

while

t E T]

is a uniform neighborhood in U*. Since f is one-to-one, we conclude that both -i f and f carry uniform neighborhoods onto uniform neighborhoods, proving that both are uniformly

continuous.

dense subspace

of

U*, where

completion

of

(V,Tu(V)).

completion

is unique

For

nat V = U*.

f

is a uniform isomorphism

the latter is complete.

Since both

up to a uniform

III.4.3 COROLLARY. if and only if

But then

(U,V)

(V,~u(V)) isomorphism a dual ~ ,

and

Thus

by Kelley V

(f,(U*,~))

(U*,~)

of

V

onto a

is a

are Hausdorff,

the

([I], p. 197).

is complete

in the U-topology

131

111.5

L I N E A R L Y C O M P A C T VECTOR SPACES

We consider here linearly and weakly topologized and linearly compact vector spaces and establish relationships III.5.1 DEFINITION. M

If

M

is a topological module if

of these notions with those already encountered.

is s module

~

(left or right) over a ring

is a topological ring, M

under addition and the action of

~

upon

M

is jointly continuous.

is a linearly topologized module if the topology of neighborhood system of

0

M

consisting of submodules.

~,

then

is a topological group Further, M

admits an open base for the A topological vector space is

weakly topologized if it has an open base for the neighborhood system of

0

con-

sisting of subspaces of finite codimension. We are interested here in Hausdorff

topological vector spaces over discrete

division rings. As a supplement

to and partly a consequence of 1.21, we have the following

result. III.5.2 PROPOSITION. (V,T)

The following conditions on a topological vector space

are equivalent. i)

ii) iii)

V

is weakly

T = Tu(V ) V

topologized.

f o r some t-subspace of

V*.

is linearly topolo$ized and every open subspace has finite codimension.

PROOF.

Items i) and ii) are equivalent by 1.21.

i) = iii).

Obviously a weak topology is linear;

the second part of iii) follows

immediately from 2.7. iii) = i).

Each subspace in an open base of the n e i g h b o r h o o d system of

hypothesis must have finite codimension, The next result further elucidates weakly topologized spaces

If

linear forms on

PROOF. V*.

By 1.17, U

Let

~

subspaces of

V

Since

0 ~ x ~ V, there exists containing function extend

x f

f

is linearly

•

D

linearly

by

V*.

is a subspace of

T

V*.

We show next that

is Hausdorff, we infer that

B t ~

such that

by letting: vf = i to all of

V.

then the set

Furthermore,

U

of

Tu(V ) ~ T

is a weak topology.

and whose intersection with on

topologized,

is a t-subspace o f

be an open base of the neighborhood system of

V.

0

is indeed weakly topologized.

the relationship between linearly and

(&,V,7)

and the equality holds if and only i f

of

V

(cf. 1.18).

III.5.3 PROPOSITION. all continuous

so

if

For any

x ~ B. B

Let

U

0

[i B = 0. D

is a t-subspace

consisting of Hence for any

be a basis of

forms a basis of

v ~ D\B, vf = 0 o ~ &, we obtain

if

B.

V

Define a

v E D [I B, and

132

of -I = iv E V I vf = o] : [y + z I Y E B, z ~

[D\B],

zf = o]

~ [D\B] I zf = O} =

= B+[z

(B + z) U z E [D\B] zf=o

which is a union of open sets and is thus open. and thus so

fu

that

U

is a t-subspace of

V*.

is continuous and 1.16 yields T

is a w e a k topology.

and 1.18, we have linear forms on

V.

T = 7uI(V) = Tu(V )

Since

Tu(V ) ~

Conversely,

T = Tu~(V )

where

U

let

U~

V/S

M,

then

PROOF.

for every

M/N

Let

# N.

If

[K~c~A

N

if

~:M - M / N

V

is weakly

0

it follows that

K s~ + N.

+N

topologized and

S

is

The next result

for linearly topologized modules.

N

in

M

III.5.5 DEFINITION. m+N

But then

0

in -I

and thus K~ ~ F~ -i P~ and hence K ~

If

N

M/N.

is closed,

m ~ K~+N.

0 Then

is a union of open sets

an open base for the neighborhood system of

$

and thus

be the canonical homomorphism.

~ an open submodule of

and since

so that

n e i g h b o r h o o d of

A family

1.21

is a closed submodule of a linearly topolo$ized

fi K ~ = 0 in M/N. ckA ~ Let P be an open neighborhood of

then the set

U = U~

is also weakly topologized.

K ~ = K

K m

m { N

N N = ~

we have

be an open base for the neighborhood system of

c~ E A, the set

Then

By 5.2,

i__sslinearly topologized in the quotient topolosY.

thus open w h i c h makes

M,

be a weak topology.

that the corresponding statement also holds

consisting of submodules and let

(m+K~)

T

xf = i u E U,

T = Tu(V ), then 5.2 implies

also has this property,

V, then

111.5.4 PROPOSITION.

m+N

If

with

for any

is the vector space of all continuous

Note that 2.12 can be rephrased thus:

module

T.

f E U

fu = u

aS required.

a closed subspsce of asserts

Consequently

With this notation,

m~ ~ K ~

M/N.

in

K~

Then

P~

-i

~ ~ A.

which proves M/N m

such that

which proves

for some

is a submodule and

is a linear variety,

Let

there exists

P 0

U (K + n ) and is hEN m E M be such that

that

is an open Since that

N ~ P~

[K TI]o~A__

-i

,

is

consisting of submodules. is an element of a module

or simply s subvariety,

of

M.

of sets has the finite intersection property if the intersection

of any finite number of members concept, due to Lefschetz

in

$

is nonempty.

We now come to a fundamental

[i]; it represents a w e a k e n i n g of the n o t i o n of com-

pactness. 111.5.6 DEFINITION.

A linearly topologized module

M

is linearly compact if

every family of closed subvsrieties w h i c h has the finite intersection property has nonempty intersection.

133 111.5.7 PROPOSITION. M,

then

M/N

PROOF.

By 5.4, M / N

closed subvarieties of ~ : M ~ M/N

If

N

intersection,

M

Let

iC }0~ A

be e family of

having the finite intersection property and let Then

tC~ ~-I~jo~A

is a family of closed

having the finite intersection property and thus has a nonempty

say

C.

~ #

Thus

c~

( fi C

=

-i)~ ~ fi c ~ -i u = fi c -- o~A ~ c~:A ~

o~A M/N

topology.

is linearly topologized. M/N

be the canonical homomorphism.

subvsrieties of

and hence

is a closed submodule of a linearly compact module

i__~slinearly compact in the quotient

is linearly compact.

III.5.8 PROPOSITION.

A discrete linearly compact vector space is finite di-

mensional. PROOF.

Let

be s basis of

V

V

be a discrete and infinite dimensional vector space.

and for every

x E B, let

S

be the subspace of

V

Let

B

generated

X

by

B\[x] .

For . any .

.Xl,X2,

,x n ~ B, we obtain

(x I + x 2 + ... + x i _ I) + x i + (xi+ 1 + ... + x n )

E x i+S

xi

n

so

i-l~(xi + S x ' )

# ~"

Since

V

is discrete,

the family

IX+Sx]x~ B

consists of

i

closed subvsrieties and has the finite intersection property. where

xi,x E B.

n m ~ ~ixi = x+j~__l~jZj i= I

Then

so that

Let

y =

~ ~ixi ~ x+S x i=l

n m ~ ~.x - ~ ~.zj i= I i i j=l j

x =

where

z

E B and x ~ z for 1 < j < m. By linear independence we must have x = x i J J -_ , for some i. It follows that y ~ x + S x for all x # x i for i < i < n and thus ~L ( X + S x )

= @.

Consequently

V

is not linearly compsct.

x6B A family if for each

~

of subsets of a uniform space

~ C ~

there exists

next result is due to Dieudonn~ III.5.9 THEOREM. PROOF.

F E ~

(X,~)

such that

is said to contain small sets

F c X[x]

for some

x E X.

The

[5].

Every linearly compact module is complete.

In view of Kelley

([i], p. 193), it suffices to show that every family

of closed sets w h i c h has the finite intersection property and contains small sets has nonempty intersection. compact module

M

neighborhood system of exists

F E ~

It follows that x E

Hence let

~

he s family of subsets of a linearly

satisfying these requirements and let

such that

0

consisting of submodules of F ~ K[x]

F ~ iY E M I x - y

for some e K]

~ M.

x ~ M~ where

which, we claim,

be an open base for the For every

K E E, there

~ = ~(y,Z) l y - z ~ K } .

is equivalent

to

N (f+K). For if F ~ [y E M I x - y % K}, then for any f ~ F, x - f 6 K so fE F x E f+K and thus x E A ( f + K ) . Conversely, if x E ~I ( f + K ) , then x ~ f + K f6F fEF

134 for every

f E

F, so

x - f 6 K

and

thus

f t ty ~ M I x - y L K]

for every

f ~ F.

Let

We have

just

Let

F+K

x+K

~

seen

~ Q

that

and

f~ F K ~ ~{, there exists

for every

x C

N

(f+K).

Then

for any

F ~ ~

such

f 6 F, we have

that x E

F+K

f+K

E ~. and thus

fEF F+K.

k' E K.

If

f C F,

For any

k 6 K, we

f+k so that

F+K~

=

If

F+K

$

has

n

closed, H 6 $

there such

x = h+k z ~ M

that

that

which

quently

Then

~ q,

that

K

F+K

is open and since = x+K

is a

then

such

that

(x+K)

For

Now

Since

N F = ~,

contradicting x ~

this

K

iet

g t H.

F

there

x 6 H+K,

implies

= h + k+ K = x+K

Therefore

6 ~

since

of closed

compactness

x %

;~ F = ¢.

H+K

m ~ H.

let that

seen above.

is a family

by linear

such

(z - h) - (z - g) 6 K

g % h +K

property.

F ~ ~

Q

which

int{,rsection;

Further

for all

and thus

Therefore

property

exists

as we have

z -m E K

H 0, Fc:_ ( x + K )

intersection

there

h ~ H, k 6 K.

implies

property.

intersection

has nonempty

K ~ }{

H+K

z - h,z - g E K

g -h E K

q

N F.

for some such

have

intersection

the finite

exists

for some

i=l

the finite

x ~

follow

i = 1,2,...,n,

~ Fi ~ ¢

the family

Suppose

for

n

implies

x = f+k'

By hypothesis

It

fi (Fi + K i) E

having

that

= x+K.

also he closed,

Fi+K i E Q

subvarieties

that

= x + ( k - k ') t x + K

i=l since

implies

then have

Therefore

it must

subvariety.

the hypothesis

(×- k')+k

x+K.

it is a submodule closed

then

exists

we have

the existence

of

In particular,

K

we

is a submodule.

proving

that

the hypothesis

is

Thus

H c

x+K.

Conse-

$

has

the finite

that

[] F.

Ft$ 111.5.10 the finite

LEMMA.

If

V

An open base

for the n e i g h b o r h o o d

[B(vl,v2,...,Vn;0,0,...,0) is linearly Let section closed

then

V*

is linearly

compact

i__~n

~

be a family of closed

subspace

Then each

of

= if( + S ~ } o ~ A , by all

j = 1,2,...,n

I vi E V}

system of

0

of the sets

each of which

consists

is a subspace

of

V*

and thus

topologized.

property.

generated

space,

topology.

PROOF.

V*

is a v e c t o r

V*.

where the

with

Letting each

SC~.

subvarieties

element

Se Let

of

~

of

V*

S : T m, we obtain is a subspace xei C Sei,

x l+Xc~2+...+x

of

f+T V.

Let

i = 1,2,...,m

= x

am

having

is of the form

+x

~l

finite where

interT

= f + T z z = f + S ±. W and

+... +x

~2

the

f+T,

.

~n

is a Hence

be the subspace x~j E S~j, Let

of

V

135 m n g E [ Q (f + S ~ )] N [ O, (f~ + S _~ )]. i=l ~i i j=l j Dj

It follows that

g ~ f

+ S t , whence (~i

~i and similarly

= f gls~j

E S~ ~i

so that gls(~ ' = f~iiS

Consequently

f + ... + x f = x g + ... + x g ~i ~i ~m ~m ~i ~m

(x i + ... + x It follows that the function +...+x

is single valued.

(~i

~j IS~j"

x = x

(x

g-f

)g = (x

f

+ ... + X ~ m ) g = x~if~l + ''" +x~mf~m"

defined on

)f = x

f

W

+...+x

by f

if

x i E S i

Further, we obtain

[Y(X~l+'''+X~m)+6(X~l+'''+X~n)Jf (~{X~l) f~l + ... + (~{X~m )f ~m + (~X~l) f~l + "'" + (6X~n) f~n = 7(xc~ifC~l+...+x = ~[(x

and hence V.

f

Then

l+...+x

is linear.

Let

f

also denote any linear extension of

f 6 V* and for every (~ E A, fls

f E f + S ±. C~

f )+6(x~if~l+...+x~ f~ ) c~m C~m Pn Pn )f] + 6 [ ( x + ...+x )f] ~m ~1 ~n

Consequently

= f 1S so (y f~ f~ (f¢ +S~).

f E

Ct

f- f

f

to all of

~ S ~,(~ that is

c~A

We may now put together some of the results on linearly compact vector spaces as follows. III.5.11 THEOREM. (g,V,T)

The followin$ conditions on a topological vector space

are equivalent.

i)

V

is linearly compact.

ii)

V

is weakly topolo$ized

iii)

V

i~s algebraically

topology, where PROOF.

U

V/K

since

K

mensional,

i) = ii).

Let

g

is also closed.

The completeness

of

isomorphic

K 6 E V

to

U*

with

K

K ~ K, in the quotient is open, and that

But then 5.8 asserts

follows from 5.9.

V/K

that

has finite codimenaion

the finite

4.

be an open base for the neighborhood

For any

is discrete since

so every

and topologically

is some right vector space over

consisting of subspaces. that

and complete.

topology of is linearly

V/K

and thus

system of

0

V/K, we have topologized

must be finite diV

is weakly topologized

136 ii) = iii). completion f

maps

of

V

By 5.2,

(V,T)

onto

U*

iii) = i).

This follows

EXERCISES.

i)

V

continuous ii)

V'

semilinear

Show

topology.

and

immediately

be weakly

that in any vector

topologized of

For information ([i], §i0);

vector

into

V~

V, the finite

V

in

on linearly

£u(V).

We consider

III.6.1 DEFINITION. VV

and Gross

of

yields

that

isomorphism.

spaces.

Show that every

is uniformly

continuous.

topology of

V*

is a weak

linear forms on

V*

V**, where

coinV**

is

of

spaces,

to

A TOPOLOGY

FOR

[5],

£u(V)

is the finite

topology of

VV

relativized

of this topology particularly

£u(V).

£u(V)

spaces

is the finite

at the beginning

£u(V)

consult Dieudonn~

[i] and the Appendix.

For a pair of dual vector

relativized

From the observations finite topology

Now the

and topological

and is dense in

a few simple properties

to the ring structure

topology of

V**

compact vector

see also Fischer

The topology we have in mind here to

V*.

topology.

III.6

relative

V

space

image of

of

from 5.10.

Also show that the space of all continuous

cides with the canonical

U

as in 4.2 which by our hypothesis

the needed algebraic

transformation

endowed with the finite

K~the

for some t-subspace

(f,(U*,~))

and provides

III.5.12 Let

T = Tu(V)

is

of Section

(U,V),

topology

the finite

~u(V)

of

£u(V).

i, we see that a base for the

is given by the sets

B (Xl,X2,.'-,Xn;Yl,Y2,...,Yn) = {a ~ ~ u ( V ) where as in 1.5, the set In particular

the sets

i = 1,2 .....

I x i a = Yi'

~Xl,X2,...,XnJ B(x;y), with

n],

can be taken to be linearly

independent.

x ~ 0, form a subbase and n

B(xl,X2,''',Xn;Yl,''',Y n) An open base for the neighborhood B

= {b E ~u(V) (Xl,X2,...,Xn)(a)

and for

system of

=

NB.x,

i=l ( i'Yi)"

a E £u(V) I xi a

is thus given by the sets

xib , i = 1,2, =

n} ...,

a = O, we have

B(xl,x2 . . . . . Xn)(O ) = (b E gu(V) / x i b = O, i = 1,2 . . . . . n},

,

137 which shows that an open base of the neighborhood system of of annihilators of all finite dimensional subspaces of

V.

0

is given by the set

Another way of ex-

pressing the above neighborhood system can be obtained as follows. 111.6.2 PROPOSITION. is minimal.

Then

PROOF. For any

Let

a ~ £u(V)

and

c =

~ [ik,~k,~k] ~ ~u(V), where k=l

n

B(Vxl'VXk'''''Vhn)(S) = {b E £u(V) I ca = cb}.

First note that

u. ,u. ,...,u i l I i2 n

are linearly independent by 1.2.5.

b E £u(V), we obtain b E B(v~I,VX2 ,..,,vXn)(a ) ~

vxib = vhia

for

i = 1,2,...,n

n k~=lUik~k(Vxk(b

- a),u)

= 0

for

all

u E U

n ~

k~lUik~k(VXk,(V= c*(b-a)*

In particular,

- a)*u)

= 0 ~

for

= 0

c(b-a)

for

= 0 ~

s = 0, we have

~u(V).

O.

We introduce some more notation.

,v

i if

IvXl,v 2 ,..,,vhn } A ring

group of

~

~

~u(V)

It follows

form an open base for the

For

a E ~u(V)

and

c E ~u(V), we let

Now 6.2 yields ,.

k2

(a)

" " 'V~n)

is a basis for

Vc, ~c(0) = ~r(C).

provided with a topology

(~,T).

III.6.3 LEMMA.

For

ca

T

is a top>logical rin~ if the additive

is a topological group and the multiplication is jointly continuous,

be denoted by

PROOF.

cb =

This gives an intrinsic characterization of the topology

tle(a ) = ib ~ ~u(V) I ca = cb}. ~ c (a) = B(v~

u E U

B(v l,Vk2 ,...,vXn)(O) = ~r(C).

that the right annihilators of elements of neighborhood system of

all

Let

d E ~c(a)

We also require that (~u(V),#u(V))

be Hauadorff.

is a topolo$ical ring.

a,b ~ ~u(V), c ~ ~u(V) and

T

to

and consider the basic open set

~c(ab).

e C flca(b), we obtain c(de) = (cd)e = (ca)e = (ca)b = c(ab)

which shows that

de ~ ~c(ab).

Hence

plication is jointly continuous.

~c(a)~ca(b) ~ ~c(ab)

~c(a) -~c(b) ~ ~c(a -b), proving that the additive group of group.

Suppose next that

a # b.

by 1.3.3 implies the existence of ca # cb; if

Then

and the topology

a -b

c E ~u(V)

d ~ ~c(a) n ~c(b), then

~c(a) ~ ~c(b) = ~

so that the multi-

A similar argument shows that ~u(V)

such that

c(a -b) # 0.

ca = cd = cb, a contradiction. ~u(V)

is a topological

is a nonzero element of

is Hausdorff.

~u(V) Hence Thus

which

138 III.6. 4 LEMMA.

In

~u(V),

we have

i)

~Z(~T(V))

= ~ ( T ~)

if

T

is a subspace o f

U,

ii)

~r(~u(S))

= ~s~(V)

if

S

i_.~s~ subspace o f

V.

PROOF. suppose

We will prove i); the proof of ii) is left as an exercise.

that

U

is a t-subspace

sequence of equivalent

(bU* ~ T

~

b E ~u(V),

v

6 V

b*U ~ T

Item i) follows

a~T(V ) = 0 ~

implies

Only the last equivalence and

V*.

from the following

statements

a E ~%(~T(V))

u.i E T

of

We may

ab = 0) ~

needs a proof.

be such that and

ab = 0

b E ~T(V)

Va ~ T m. bU* ~ T

Suppose =

(v~,u i)

b*u i = u i.

for all

V # 0

Hence for any

implies

and let v ~ V

ab = 0.

Let

b = [i,y-i,x]. using

Then

the hypothesis,

we obtain (va,ui) = (va,b*ui) and since

= (v,(ab)*ui)

= (v,O) = 0

u. % T is arbitrary, we have (vast) = 0 for all t E T i Conversely, suppose that Va ~ T ~. Then for any b E £u(V)

Va ~ T ±.

b*U ~ T, we immediately since

= (v,a*b*ui)

b*u E T

and

obtain

Va c T m.

III.6.5 COROLLARY. we have in

for any Thus

For any

u E U, v E V, (v(ab),u)

v(ab) = 0

c E gu(V)

for all

and

v E V

so that for which

= (va,b*u)

= 0

and hence

ab = 0.

[Vl,V2,...,Vn]

a basis of

Vc,

£u(V),

~(vc)~(v) = ~r(~(vc)) = B(v l,v 2,...,vn)(O) = ~c(0) = ~r(C). Note

that in view of 2.7,

a base for the neighborhood

the family of all open subspaces

system of

0; an analogous

situation

of

V

constitutes

occurs

in

~u(V),

viz. III.6.6 COROLLARY. of

~u(V)

constitutes

topology

T

of

U

V.

~T(V)

IT

open subspace

of

UJ

system of

of right i d e a l 0

for the

#u(V).

PROOF.

of

The family

an open base for the neighborhood

Interchanging

the roles of

is open if and only if For each such

from 1.2.3-2.5. the family

~r(C)

S

T = S~

there exists

Hence by 6.5, where

U

c

and

V

is 2.7, we infer that a subspace

for some finite dimensional c ~ ~u(V)

such that

the family in the present

varies over

Vc = S

corollary

subspace

S

as follows

coincides

with

~u(V), which by the above discussion

has

the required properties. III.6.7 EXERCISES. i)

Find necessary

and sufficient

conditions

in order that

~u(V)

be discrete.

139 ii) S

of

For a dual pair

~u(S),

and finite dimensional ~u(V)

subspaees

T

relativized

to

U

and

~T(V)

of

and

respectively?

iii)

Let

bilinear

(U,V)

satisfy all the conditions

form may be degenerate•

the bilinear iv) T~ =

(U,V)

V, what can be said about the topology

Let N

~r(v) V)

What are necessary

form in order that (U,V)

of a dual pair except

~u(V)

be a dual pair.

and sufficient

that the

conditions

on

be Hausdorff. Show that for any subspace

T

of

U, we have

Na .

Show that

(~u(V),$u(V))_

has no proper closed ideals.

For the origin of the results of this section as well as for further discussion, see Jacobson

[5],([6], Chapter

IX, Section 6),([7], Chapter

111.7 The topology in question

is the topology

henceforth will not be mentioned properties or

U

of one-sided

~.

there exists

Let

for some

#u(V)

relativized

to

~u(V)

and

We study here a few topological

The closure of s subset

c E ~u(V).

such that

u.1 6 U, we have

a 6 ~(S).

left ideal of

and

Consequently

(~v)a and thus

~u(V)

The next two propositions

Eyery

a E ~u(S)

b E ~(S)

= Veb ~ vb ~ S.

~u(V).

18).

A

of

~u(V)

• t are due to D1eudonne

[2],

[5]•

III.7.1 PROPOSITION. PROOF.

explicitly.

ideals of

will be denoted by

see also Jacobson

A TOPOLOGY FOR

IV, Section

cb = ca.

Vca ~ S

Hence

III.7.2 PROPOSITION.

~(S)

is closed.

hc(a ) Fi ~ ( S )

Hence

for all

(vh,ui) = ~ ~ 0 = ~vk[i,o-l,~]a

~u(V) Then

Vb ~ S

c E ~u(V).

# ~

and hence

so that Let

~v

Vca ~ V; then

so that

E S

c ~u(S).

For every right ideal

R

o__ff ~ = ~u(V), we have

= ~r,$~%,~(R). PROOF.

First note that

~r,~,$(~T(V))

Let

= ~T ~m(V)

= ~ (V) by 6.4 and 2.2. T independent vectors in V. Then

a 6 ~_(V) and Xl,X2,...,x n be linearly T s*U ~ ~ and letting S = [Xl,X2,...,Xn], for any

u 6 U, we have

Further, we can write

minimal.

a =

~ ' k=l [~k'Vk'kk ]

with

m

( a * u + S ~) A T # @.

It follows

that

m (k~=lUik~k(Vkk,U) +S±) •

By I 2.3,

there exist

t

/

Ul u2''"

.

u

!

N T # ~

' m £ U

such that

(u 6 U). (vAi,u!)j

=

6iJ

for

1 _< i,j _< m

140 since infer

V%l,VX2,...,VXm

are linearly

of

the existence

g k E S±

independent

such

that

by 1.2.5.

Computing

t k = U i k ' Y k + g k E T.

a Uk, we

Letting

m t k = u .Jk "rk

and

b ~ ~ l[Jk,Tk,Xk] " , we obtain

m ~ u. T. (v

b*u =

k=l

]k k

m ~ t (v

,u) = )'k

k= 1 k

,u) E T )~k

and m

Xpb = k~__l(Xp,Ujk)TkV x k = k~= l (Xp ' t k ) v/"k m ~ (x ,u.

=

k=l for

1 < p < n

k ~'k

B(Xl,X2,.-.,x n)

(a) fi ~T(V)

(1)

a E ~T(V).

Conversely, vectors

= Xpa

)~.v

~k

so that

b E and thus

P

in

V

Xkb = xka

let and

for

a E ~T(V).

Let

u ~ U, Xl,X2,...,x n

S = [Xl,X2,...,Xn].

1 ~ k < n

and

There

b*u ~ T.

Let

exists

be linearly

b

t = b'u;

satisfying then

independent

(i),

so

t = a ' u + (b ~ a)*u,

where (Xk,(b - a)*u) = (Xk(b - a),u) = (0,u) = 0. Thus

t C

( a * u + S m) N T

which

proves

that

a*u E ~

and hence

a*U ~ T,

that is

a E g_(V). T III.7.3

COROLLARY.

For any subspace

gT(V)

= ~_(V) T

It is clear that the set is always

a right

generated III.7.4

by

ideal,

= ~ ~ T

of

U, we have

(V) = ~ , ~ , ~ ( a T ( V ) ) .

~r,~(A),

and that

T

where

~r,~(A)

A

is a nonempty

= ~r,~(L),

where

L

subset

of a ring

A.

COROLLARY

c(~.

2.5).

Th____eefollowing

conditions

on a subspace

T

of

are equivalent. i)

T

is closed.

ii)

~T(V)

is closed

iii)

~T(V)

is ~ right

PROOF.

This corollary subspaces

of

corollary

is valid

111.7.5

annihilator.

Exercise.

U

~,

is the left ideal of

implies

onto closed

that the lattice right

ideals

for open subspaces,

COROLLARY.

isomorphism

of

Su(V).

see 7.10,

Every principal

risht

X

in 1.3.10 maps

A slight variation

exercise

ideal of

iv). Su(V)

closed

of this

is closed.

U

141 PROOF.

Exercise.

An important kind of right ideal is provided by the following. III.7.6 DEFINITION.

Let

R

be a right ideal of a ring

is a modular

left identity of

~

relative to

such a case

R

R

if

~.

r - ar E R

An element for all

a E

r ~ ~; in

is a m o d u l a r risht ideal of

The next three results are new. 111.7.7 PROPOSITION. if

T

A right ideal

~T(V)

o_~f ~u(V)

contains an open subspace. PROOF.

r E ~u(V). u = uto.

Necessity. Then

s =

For every

By hypothesis

r - ar E ~T(V)

n

There exist

and

a E ~u(V)

and all

u E

and

(Va) ~

n

Vl,V2,...,v n E V

vk

Let

(k~=l(V,Uik)NkV~k,Ut) = k=l ~ (v,ulk)~k(VXk,Ut) . such that

(v.,u. ) = ~jk J ~k

1.2.3, and hence the last equation implies ~

for some

~ [ik,Nk,~k] , w h e r e n is minimal. k=l v E V, we have (va,u) = 0 and thus

0 = (va,ut)=

Let

is modular if and only

be such that

that

N ( v k , u t) = o.

u = ut~ = ut~(v

for

i ~ j,k < n ---

(Vkk,Ut) = 0

for

by

k = 1,2,...,n.

Then

,ut) n

= [t,~,k]u t - ( ~ [ik,~k,kk])[t,~,~.]u t E T k =i = by hypothesis.

Consequently

Sufficiency.

Since the subspaces

the n e i g h b o r h o o d system of

dimensional subspace Ve = S

implies

S

For any

V. Thus

dim Va < oo

S ~, where

0, the hypothesis

of

(e.g., use 1.2.3). u E T.

(Va) ¢ c_ T, where

implies that

(Ve) ¢ C T, that is and

(Va) m

is o p e n

dim S < ~, form an open base for S mC

There exists an idempotent

v E V, u E U

and thus

T

for some finite

e ~ ~u(V)

(ve,u) = 0

such that

for all

v E V

r C Su(V), we obtain

(ve,(r - er)*u) = (ve,r*u - e*r*u) = (ve,r*u) - (ve,e*r*u) = (ve,r*u) - (ve,r*u) = 0 so that

(r - er)*U c_ T

III.7.8 COROLLARY.

and therefore ~u(V)

r - er E ~T(V).

has an idempotent m o d u l a r

left identity relative

to every m o d u l a r right ideal. PROOF.

This follows from the last part of the proof of 7.7.

111.7.9 T H E O R E M (cf. 2.9).

The followin$ conditions on a subspace

are equivalent. i)

T

is a closed hyperplane.

T

of

U

142 ii)

~T(V)

is a closed maximal risht ideal.

iii)

~T(V)

is a modular maximal risht ideal.

PROOF.

i) ~

i) = iii).

This follows from 7.4 and 1.3.4.

This follows from 7.7 in view of 2.9. n Let e = ~ [ik,~k,X k] E ~u(V) be such that k=l

iii) ~ i). r E ~u(V).

all

ii).

Let

u i E U\T

plane, it follows that

and

r = [j,6,v].

u. = t + u . T J i

for some

Since

t E T

T

and

r-er is

E ~(V)

evidently

T E A.

for

a hyper-

Hence

(r - er)*u = ([j,6,~] - k=l~ [ik,~k(Vxk,Uj)~,v])*u n - ~ u. ?k(V. , u . ) ) ~ ( v ,u) (uj k=l ik ~k J n n = I t - k=l~U.lk~,(vmA k ' t ) - ['~ luU iik ~ )k ( kV X k= for all

t E T, • E A, u E U. n

-ui]T}6(v~'u)

E T

Consequently n

u i Vk(V I ,t) + [ ~ u i Vk(V k ,ui) - ui]T E T k=l k "k k=l k k for all

t E T, T E A.

For

t = 0

and

T ~ 0, we obtain

n

(1)

k~=lUik~k(Vkk,Ui ) - u i E T and for

T = 0, n

k=l

(t E T).

u i Vk(V k ,t) E T k k

By 1.14, we have Hence

(1) yields

T = gZ for some g 6 U* relative n g ( ~ u. ~ , ( v ,u.) - ui) = 0 whence k= I i k K Ak i n gu i =

and b y ( 2 ) ,

(2) to the dual pair

(U,U*).

(3)

~ (gu i )~k(Vk ,ui), k=l k k

we o b t a i n n

(4)

(gu i )~k(Vk ,t) = O. Now s u p p o s e and

(4)

that

k=l

k

(Vxk,U)

= 0

k for

k = 1,2,...,n.

Then

u

gives gu = g(t +uiT ) = gt + (gui)T

= k=if(guik)Vk(Vkk,Ui,) = k=l~(guik)Vk(Vxk,U)

= (gui)T n

= k~=l(gUik)Vk(Vkk,U - t)

= O.

=

which by (3)

t+u.T i

143

It follows

that for

S = [fvxl'fvx2'''"fVkn]

g E by 2.3.

Consequently

T = g~ = v z

so that

S ± ~ gZ

[g] = g± ~ ~ S ~ ~ = S

g E nat V

and hence

, we have

T

and thus

is open since

g = fv

for some

v E V.

v ~ = fvlO, and thus

T

But then

is also closed.

III.7.10 EXERCISES. i)

Show that a maximal

ii) if

V

right ideal of

Prove that every right ideal of

~u(V)

~(V)

is either

closed or dense.

is a right annihilator

if and only

is finite dimensional.

iii)

Show that for any right ideal

R

of

Su(V),

we have

= ~a E ~u(V) I n N r c_ Naj. rER iv) of

For a dual pair U

v)

(U,V),

prove that

the following

conditions

on a subspace

are equivalent. a)

T

is open,

b)

~T(V)

is open.

c)

~T(V)

is the right annihilator

Show that in

~(V),

provided with

of some element

the finite

of

topology,

~u(V). every left ideal is

a left annihilator. vi)

In any ring

~

(not necessarily

(i - s ) ~

~r - ar

~}

Ir E

a E ~. Show that for e = e , we have (i - e)~ = ~r(e). 2 where 6 ~u(V), we have ( i - e)~u(V ) = $ ( l _ e ) , u ( V )

Also show thst

e = e

(i - e ) * U = ~ u - e*u l u E U} relative vii)

to

~T(V)

For any

viii)

Let

a E ~u(V)

The principal

we

let

Mr(S ) T

left identity

=

~(Va)X

for

~u(V)

(v).

be a subspsce

~T(V) = ~r(S)

for some

of

U.

Show that

a E Su(V).

for this section are Behrens [5],([6], Chapter

([i], Chapter

IX, Section 8),([7], Chapter

II, Section I, Section 3

18). III.8

~,

is a modular

(Ve) ± ~ T.

show that

references

[2], Jacobson

IV, Section

For a set

e

be a dual pair and

if and only if

7), Dieudonn~ and Chapter

and that

if and only if

(U,V)

codim T < ~

E

write

2

for any for

=

with identity),

of functions

ANOTHER TOPOLOGY on a set

X

FOR

£u(V)

into a uniform space

(Y,~),

for every

144

W(~) : [(f,g) E $ x $ Then the family uniformity

I (f(x),g(x)) E ~

[W(~) I ~ E ~}

of uniform

is a base

convergence;

for all

x E X].

for a uniformity

the corresponding

on

topology

$

called the

is the topology of

uniform convergence. We now apply this construction to the set i.

£u(V)

of functions

An open base

family

[T~ I T

2.

v E A

The uniformity

V 3.

for which According

convergence where

there exists

associated with we proceed

system of

U, dim T < ~}

~

on

V

0

in

Tu(V)

and

as follows.

V

is given by the

so that a set

A c V

a finite dimensional

induced by

for

on

Tu(V )

is a ~u(V)-neighborhood

~(A) = {(x,y) Ix - y ~ A].

on

Specifically

is open if

subspace

T

of

U

v + T z c A.

[~(A) I A where

V.

for the neighborhood

is a subspace of

and only if for every such thst

to the uniformity

on

Consequently

there exists

£u(V)

of ~

a neighborhood

to the above construction, induced by

~

has for a base the family 0}

consists A

of

0

of all binary relations such that

the uniformity

has as a base

~

~(A) ~

~.

of uniform

the family

[W(~) I ~ E ~ ,

~ = ~(A),

~('L((A)) = [ ( a , b ) E £u(V))<~,u(V ) /V(a - b ) ~_ A}. 4.

For any neighborhood W(~(A))[a]

A

of

= [b E £u(V)

and thus the topology of uniform consists

of those sets

neighborhood

A

of

C

0

0

with

and

a E ~u(V),

IV(a-b)

convergence

such that

£u(V)

(simply uniform

that for every

W(~(A))[s]

For every finite dimensional

~ A}

on

the property

we have

subspace

topology)

a E C, there exists a

c C.

T

of

U, let

= ~a E ~u(V) I Va ~ T ~} . 111.8.1

PROPOSITION.

The family

an o~en base of the neighborhood PROOF. have

Let

Vs c T ~

T

iT IT

system of

is a subspace o f 0

be a finite dimensional

U, di___~mT < ~}

for the uniform subspace

of

U.

topology For any

on

forms

~u(V).

a E T, we

so that

W(~(TZ))[a]

= [b E £u(V) I V(a-b) ~ r ~} = [b E £u(V)

I -vb E - v a + T ±

for all

v E V}

= [b E ~u(V) I Vb ~ T m} = and thus for every Since

A

T

is open.

Next

let

a E C, there exists is a neighborhood

of

C

be an open set in the uniform

a neighborhood 0, there exists

A

of

0

topology.

such that

a finite dimensional

Then

W(~(A))[a] ~ C. subspace

T

of

145

U

such that

T ~ c A.

Hence

a + T = ~ a + b Ib E ~u(V), Vb 6- T ±} = ic E £u(V) IV(c- a) 6_ T -~} = [c E ~u(V) Iv(a-c) 6_ T ~] = W(~(T±))[a] ~ C as required. III.8.2 COROLLARY.

The family of sets a)

P(ul,u2,...,Un)( where

{Ul,U2,...,Un]

constitutes

ranses

an open base

{b E

=

Ib*u k

~u(V)

over all linearly

for the neighborhood

a*u k

=

for --

independent

system of

k = 1,2, finite

..

.,n}

subsets

a E ~u(V)

of

U,

in the uniform

topology. PROOF.

It suffices

~Ul,U2,...,Un}

to show that

is a basis of

b E a+T

T.

~=~ b = a + c ,

(v(b-a),uk)

= 0

(v,(b*-a*)uk) b*u k

a*u k

vc E T m for all

= 0 for

i__~sminimal, PROOF.

v E V

for all

and

c E ~u(V),

PROPOSITION. we have

where

v E V

k = 1,2,...,n

and

.,n ~

and all

k = 1,2,...,n

b E

(a) P(ul,u2 ,..,,u n)

"

Me(a ) = {b E £u(V) I bc = ac]. n a E £u(V) end c = ~ [ik,~k,X k] E ~u(V), k=l

For

let

where

n

[~c(a) = P(Uil'Ui2'' "''Uin)(a)"

Note that

b E £u(V),

c E £u(V)

and

v E V

k = 1,2,

v

,v ~i

any

for some

. .

a E £u(V)

III.8.3

= p(u I 'u2'''" ,Un)(a),

We have

=

For

a+T

,...,v X ~2

are linearly

independent

by 1.2.5.

For

n

we obtain

b ~ P(Uil,Ui2 ' ...,ui

)(a)

~

b*u k = a~'~uk

for

k = 1,2,...,n

n

n k=l~ ( v , ( b * - a * ) u k ) ~ k V ~ k ~= ~ (v(b - a),Uk)~kVkk k=l ~= v ( b - a)c = 0 bc = ac ~ In the special

case

for all

= 0 = 0

for all for all

hood system of

= 0

b E ~c(a). a = 0, we have

the uniform 0

v ~ V

v E V ¢~ (b -a)c

P(Uil,Ui2,...,Uin)(0 ) = ~c(0) Consequently

v ~ V

consisting

topology

on

= ~(c).

£u(V)

of left annihilators

has an open base of elements

of

for the neighbor~u(V).

Since

the

146 corresponding annihilators immediately

statement

of

Su(V),

is intrinsic

for the finite

topology

it follows

obtain all the statements

sponding ones

nonzero

for the finite

of elements

£u(V)

for the uniform

topology.

and can be defined

for

topology

topologies

on

£u(V)

of the socle.

have been defined

they turned out to be symmetric these remarks,

system of

to the left-right

that the following

we

topology

ring with a

0

It is interesting

in quite different ways,

relative

it is not surprising

duality,

from the corre-

simply as a topology on a primitive

of elements

for right

This also shows that the uniform

socle having an open base for the neighborhood

of left annihilators

is valid

that by the left-right

consisting

that these two but,

duality.

connection

in the end, In view of

between

the two

is valid. III.8.4 PROPOSITION. homeomorphism o f topolosY

£u(V)

with

a - a*

the finite

induced by the V - t o p o l o g y on

PROOF. ~v(U),

The mapping

Further,

~c(a)m

for

topolosy onto

isomorphism

£v(U)

with

and hence

the uniform

U.

We know that the mapping

see 1.1.2.

is a uniform

~:a - a*

a E ~u(V)

is an isomorphism

and

c E SU(V),

of

~u(V)

onto

we have

ca]m

= {h ~ £u(V) feb =

= [ 6* ~ £V (U) I c'b* = c'a*] = ~Lc.(a* ) which proves

that

~

is also a homeomorphism.

Furthermore,

we have

%(mr(C))m = ~(x,y) Ix-y E ~r(C)]m = [(x~,ym) Ix-y ~ ~r(C)] = ~(z~p,y~) I xR0 -yq0 E ~r(C~)} which shows

that

~

is also a uniform

Some of the results Section

18), others

If we consider

COMPLETE

a primitive

ring

~

~u(V) ~ ~ c ~u(V)

on

£u(V)

questions:

(i) What are the completions

the answer

We consider

to the first question,

111.9.1 LEMMA. £(V)

IV,

If

£(V)

i__~sdense i f and only i f

RINGS

with a nonzero

on

and sufficient

U

of

~, and since

V*.

Hence

they are

we may ask the following

of the uniform

first the finite

spaces

conditions

induced by these for completeness

topology on

£u(V)

of

and using

reply to the second.

i__~s$iven S

socle as a ring of linear

for some t-subspace

for some uniformities,

(2) What are necessary spaces?

PRIMITIVE

two topologies

uniform

topologies?

topologies

induce

necessarily

these uniform

([7}, Chapter

are new.

then

the two topologies

isomorphism.

in this section can be found in Jacobson

111.9

transformations,

= ~(~r(C~))

the finite

is n-fold

topology,

transitive

then a subset

for ever7 positive

S

of

integer n.

147 PROOF.

This follows easily from the description

of s base of the finite

topology in Section 6. Note that 9.1 says that the "density" density in the finite topology of III.9.2 PROPOSITION. PROOF. considers of

V*

~(V)

defined in 1.2.9 coincides with the

~(V). i_!s complete

in the finite topolosy.

The reasoning here is quite analogous ~(V)

as a subset of

as a subset of

V V, where

AV, where

A

V

to that in 4.1; indeed, one

has the discrete

has the discrete

topology,

topology.

instead

The details of

the proof are left as an exercise. The next result is due to Dieudonn~ III.9.3 COROLLARY. t-subspace i) ii)

U

of

Let

V*.

(a ~ e*,

ii)

i)

(~(V),~))

(~(U),~)) By 9.1, ~

The mapping

morphism of

~u(V)

be a rin$ such that

is the completion o__~f ~

is the completion o__~f ~ is dense in

~:a ~ a*

in

Hence

~

and

(~,(~(U),~)),

(£(U),~)

in the uniform topology.

Sv(U) ~

~v(U)

~(U)

is complete in the finite so by 9.1, ~

~

is dense

in the uniform topology.

and thus this completion

is unique up to a

([i], p. 197).

We have seen in Sections 6 and 8 that both topologies on ~ In view of this,

and uniform iso-

with the finite topology

~ ~v(U),

must be a completion of

are Hausdorff,

uniform isomorphism by Kelley

in the finite topology.

is an algebraic

The dual of 9.2 shows that

On the other hand, we have

Both

for some

which is complete by 9.2.

topology onto

topology. £(U).

(~(V),~)

(a E £u(V))

with the uniform

by the dual of 8,4.

~u(V) ~ ~ ~ ~u(V)

Then

(identity map,

PROOF.

~

[2].

it makes sense to introduce

the following

as in 9.3 are intrinsic.

two topologies

directly

on an abstract ring. III.9.4 DEFINITION. topology on

~

[~r(C) I c E ~}

Let

~

be a primitive

is the risht socle topology

the topology on

is the left socle topology for In light of the results is a uniform isomorphism, topology,

topology.

~; dually, 0

the family

The

the family

{~(c)

I c~]

~.

in Sections

6 and 8, the canonical

and thus also a homeomorphism

onto a subring of ~

for

system of

0

~.

system of

having as an open base for the neighborhood

well as of

ring with a nonzero socle

having as an open base for the neighborhood

~u(V)

containing

of

isomorphism

~ - £(V)

~, with the right socle

~u(V), with the finite topology,

as

with the left socle topology onto the same subring with the uniform

The above corollary

thus yields

148 III.9.5 COROLLARY. ~:~ ~ £u(V) i)

Let

(~,(~(V),~))

ii)

(~(a ~ a*),(~(U,~)))

III.9.6 COROLLARY.

A primitive

~

PROOF. second)

onto

~

into

a left primitive ~(V)

carries

for any element N

in the left socle topology.

with ~ nonzero socle is complete

follows immediately

V.

~

i__!sisomorphic

Every algebraic

in t_~o

is___~o-

from the first (respectively

statement

can also be proved directly

for we know that in view of the hypothesis, left ideals.

(respectively

~(c))

in the socle of

~

onto the uniform neighborhoods

isomorphism

~

~

right) vector space

ring with minimal

c

ring

Note that the bracketed £v(U),

~r(C)

in the right socle topology,

is uniform and hence a homeomorphism.

The first statement

part of 9.5.

by mapping

~

left) socle topology if and only if

~(V)

and let

Then

i_!s the completion of

for some left (respectivel~

morphism of

in

be a primitive ring with a nonzero socle,

isomorphism.

is the completion of

the right (respectively ~(V)

~

be the canonical

Any isomorphism

onto

~r(C~)

and thus carries in

£(V).

Hence

~

~

of

is also ~

(respectively

onto ~(c~))

the uniform neighborhoods ~

must be a uniform

and thus also a homeomorphism.

We will now derive a topology for a semiprime extension of its nonzero socle.

The procedure

prime rings with a nonzero socle whose as linear transformations

ring which is an essential

is somewhat

similar to that for

topology was derived

on a vector space.

from its representation

We will now go via a complete direct

sum. III.9.7 LEMMA.

Let

[~¢~A

be a family of prime rings with a nonzero socle

each endowed with the right socle topology. the product

topology has the famil X

neighborhood

and

system o f

O, where

PROOF.

Elements of

a E ~.

Then there exist

i = 1,2,...,n

and

~

c~ = 0

P~ = Ib E N I b~ ~ P~}.

~

Then the ring

i~r(C) I c ~ ~}

~ ~ endowed with o~fA ~ as an open base for the

is the socle of

~.

will be written in the form ~l,~2,...,~n ~ A

otherwise.

By definition

forms a subbase for the product

~ =

a = (a).

such that

c i ~ ~

For every open set the family of sets

topology

in

~.

Let

P~ ~

in

c ~

for ~,

I ~E A, P~

let open in ~ ]

With the notation just introduced,

we assert

a+~r(C) For let with

b E a +~r(C).

c~i d ~i = 0

b = a+ d

Then

and thus

(1)

= i=l~s~i +~r(C~i)

b

where

cd = 0.

E a(~i + ~/r(cc~i) •

Hence

Further,

b

= a

+d

a + ~ r ( C ) ~ a i +~r(C~i )

149

and thus also Then

b~i = a ~ i + d ~ i ,

# ~i

~

~ a +~r(C). i=l ~i "

where

[I a +~r(C i=l ~i

Now if exists

i"

c~ ~ ~

P~

Furthermore,

product

(i) shows

is an open set in ~-- P~"

~

Now let socle

~.

~

be a semiprime

By 11.5.9,

and an isomorphism contains

system of

~

Conse-

and there

for which

c

= 0

topology.

forms a base for the

{~r(C) I c ~ ~}

forms an open

O.

~

extension of its nonzero

, a E A, with nonzero socles

into a complete direct sum

N ~

such that

~ ~ . Giving each ~ the right socle topology and c~:A ~ topology, ~ can be given a topology for which ~ is a homeo~.

In particular, ~0

has the relativized

and thus has an open base for the neighborhood annihilators

system of

of the elements of the socle by 9.7.

which shows that the topology is intrinsic, representation

~.

~.

0

Let

~

The topology on

This property

the family

topology

of right

carries over to

i.e., does not depend upon the particular

be a semiprime ~

product

consisting

It is clear that we can proceed directly

III.9.8 DEFINITION. nonzero socle 0

if

the socle of

n ~ the product c~A ~ morphism of ~ onto

system of

~ P~

is open in the product

ring which is an essential

mapping

a

c 6 •

I a 6 ~, c ~ ~}

there exist prime rings

~

d~ = b~ - a~

(i).

~ , then

and thus the family of sets

base for the neighborhood

let

) ~ P~.

a +~r(C)

[a+~r(C)

n ~ i=l ~i +~r(c~i)'

bE

cd = (c d ) = 0.

Hence for

= a +~r(C

that each

the family of sets

topology of

with

which establishes

~ # ~, we obtain by (i), a + ~ r ( C )

let

i = 1,2,...,n;

b = a+d

a + ~ r (c)

such that

Conversely, for

We obtain

) ~ a+~r(C) --

a 6 P~, where

Consequently

~

c~id~i = 0

fOrn i = 1,2,...,n.

quently

if

a+~r(C)

as follows

ring essential

(cf. 9.4).

extension of its

having as an open base for the neighborhood

{~r(C) Ic ~ ~}

is the right socle topology

for

~.

The

left socle topolog~ is defined dually. The following result is new. 111.9.9 THEOREM. of it___ssnonzero socle containing For each containing (a)~

Let ~.

the socle of ~ E A, .let ~U ( V )

= (a ~ ).

Let

~

~ Let

be a semiprime ~

~ ~ , where each

~

be the ~

is an essential

(U ,V ).

~

extension

~ ~A ~ is a prime ring with a nonzero socle.

be an isomorphism __°f ~

for some dual pair T

ring which

be an isomorphism of

onto a subring of

onto a subring Define

socle topology on

~

~J~

__°f £U ( V )

on

~ ~ by 0~A ~ ~, give each £(V )

the

150 finite

topology

(q09,( I~ ~ ( V

and let

),@))

~

be the ~roduct

is the completion o f

topology on

~ ~(V ¢6A

).

Then

(~,~).

~A PROOF. of

9

First note that the existence

by II •2.8 and 1.2.13.

uniform isomorphism the product cussion

~

it.

~p

a uniform

By Kelley

of

q0

is guaranteed

by II.5.9

the right socle topology,

onto a dense subspace

topology makes

following

Giving

of

~(V )

by 9.5.

isomorphism

([I], p. 183),

~

9~

and that

becomes

Further,

a

giving

II

in view of 9.7 and the dis-

is a uniform

isomorphism

of

II

onto a subring of ring of

onto a subII ~ ( V ). Hence q0~ is a uniform isomorphism of ~ ~A ). In view of 9.2, each £(V ) is complete so that by Kelley ([i],

II £(V

~6A p. 194),

II ~ ( V

)

is complete.

It is easy to see that

~

maps

the socle of

~A onto the socle of

~I ~ ( V

).

By 9.1 and

i.2.13, we know that the socle

gu

(v)

of

~A

~ ~

is dense in

exists

~(V

c i E Pi [~ ~U

)• (V

c~. l

a direct that

for

is an open subset of

i = 1,2,...,n.

Since

~(V

), then there

the socle of

II ~ ( V

)

is

c~:A

sum of the socles of all

£(V

c~i = c.i

for

), as it easily i = 1,2,...,n

follows

from II.5.8, we have

c~ = 0

and

otherwise,

is an

of the socle of

II ~ ( V o~A is also an element of

c

that

)

Pi

i

c = (c~), where

element

Hence if

dense in

~I ~(V )

). With the notation of the proof of 9.7, we have n n P.. This shows that the socle of ~I £(V ) is i= 1 i ~A which completes the proof of the theorem•

~A III.9.10 COROLLARY. topology i f a n d only i f

A ring

~

as in 9•9 i_~s complete

~

is isomorphic

all linear transformations

on left vector

in the right socle

to a complete

direct

spaces,

~ ~(V

say

sum of the rings o f ).

Furthermore,

if

a6A both

~

and

~ £(V

)

are provided with

the topologies

as in 9.9,

then every

~A algebraic

isomorphism

of

~

onto

II ~ ( V

)

is a uniform

isomorphism

and hence a

o~A homeormorphism. PROOF.

Exercise.

III.9.11 EXERCISES. i) family

Let

~

be a ring satisfying

{~r(C) Ic E ~}

satisfies

the hypothesis

the requirements

in 9.9.

for the neighborhood

system of

0

there exists

such that

~r(C) ~ ~r(a) ~ ~r(b).

ii)

c E ~

What are necessary

hypothesis topologies?

in 9.9 in order

for a topology on

and sufficient that

~

conditions

be complete

Show directly

that the

on a family of sets to be s base ~, i.e.,

on a ring

that for any

~

a,b E ~,

satisfying

the

both in the left and the right socle

151 iii)

Let

~

be a semiprime ring with a nonzero socle

Show that the family

[~r(F) IF ~ ~ + ~

and

F

is finite}

quirements for a base of the neighborhood system of topology.

Show that

essential extension of

(~,T)

found in Leptin

0.

and let

~, how does

T

~ = ~r,~(~).

satisfies the re-

Let

is a Hausdorff topological ring.

A characterization of

Dieudonn~

~

T

denote this If

~

is also an

compare with the right socle topology of

2?

~I ~(V ) as a semisimple linearly compact ring can be c~A ([2], Part I). For more information on complete rings, see

[2],[3]

and Eckstein

[I].

152

APPENDIX

ON LINEARLY COMPACT PRIMITIVE AND SEMISIMPLE RINGS by Richard Wiegandt 0.

INTRODUCTION

In ring theory the celebrsted Wedderburn-Artin significance.

It states that a ring

~

(in this case semisimple means semiprime)

Structure Theorem is of central

with d.c.c, on right ideals is semisimple if and only if

~

is s finite direct sum

of rings of linear transformations on finite dimensional vector spaces over division rings.

It was Leptin

[2] who succeeded in eliminating both finiteness conditions

from this characterization;

he proved that linearly compact semisimple rings are

just complete direct sums of rings of linear transformations on vector spaces over division rings.

The price he had to pay was topological,

mathematician may object that linear compactness conditions.

Nevertheless,

Leptin's result is in a certain sense the best possible

generalization of the Wedderburn-Artin Structure Theorem. rings started, Zelinsky).

and a topologically minded

is actually a kind of finiteness

The study of topological

in fact, earlier (see the cited papers of Kaplansky, Dieudonn~ and

In particular,

[3] and Zelinsky

Wedderburn-Artin-like

theorems were obtained by Ksplansky

[2], but these theorems are consequences of Leptin's result.

It is surprizing that in the flood of mathematical publications only some dozen papers dealing with linearly compact rings have appeared in the last fifteen years since the publication of Leptin's papers.

What is even more surprizing, Leptin's

already classic results are not available in any textbook, monograph or even in lecture notes.

This strange situation may be due to several reasons, one of them is

definitely the fact that the study of linearly compact rings requires certain familiarity both with algebra and topology. It is the purpose of this appendix to discuss Leptin's results concerning linearly compact semisimple rings and several related topics.

The preceding lecture

notes by Professor M. Petrich, particularly Part III, provide an excellent and natural background for this aim. the preceding lecture notes,

Taking into consideration the material presented in the discussion of linearly compact semisimple rings can

be approached easily via linearly compact primitive rings.

This method differs from

153 that of Leptin, Moreover,

so there

our approach

the use of inverse Of course, explained

Finally,

of the theory

appendix.

I express

my

this

to describe

extensive

knowledge

zero socle were

of linearly

further

gratitude to his

results

proofs.

topological;

compact

however,

rings

and recent

to Professor

lecture

notes

can be

developments,

we

M.

Petrich

for the oppor-

and for his critical

remarks

of p r i m i t i v e in what

the following

and II.l.21.

RINGS

of linearly

rings.

So

follows we

and faithful ~ - m o d u l e s

in II.l.20

i.i LEMMA.

MORE ABOUT PRIMITIVE

the structure

considered,

Irreducible

have

the original

this appendix.

In order

defined

For

sincere

appendix

i.

more

with

and less

at the end of this note.

tunity of a t t a c h i n g concerning

algebraic

inevitable.

fragments

in this short

a few coincidences

to be more

limits was

only

give references

are only

seems

far only

shall not

as well

Concerning

compact

primitive

primitive impose

as primitive

irreducible

rings,

we need

rings w i t h non-

this requirement.

rings

have already

and faithful ~ - m o d u l e s

been we

simple

If

M

is an irreducible

faithful ~ - m o d u l e

and

0 # x E M,

then

x~ = M. PROOF.

or

Since

x~ = M.

x9~ ~ M,

Suppose

~

the i r r e d u c i b i l i t y

= 0

and consider

of

M

implies

that either

x~ = 0

the submodule

<x) = {kx I k = 0,+i,+2 .... }. The

irreducibility

0 = <x>~ = ~ ,

of

M

implies

contradicting

<x~ = M,

the assumption

further that

since

M

x~ = 0, we obtain

is faithful.

Hence

~

= M,

as asserted. 1.2 LEMMA. ~(h,V)

of all

Moreover,

if

linear M

over a suitable PROOF.

Every

primitive

transformations

is a faithful division

We have

tins

ring

~

i__~s isomorphic

on a v e c t o r

to a subrin$ o f

space

irreducible ~ - m o d u l e ,

V

then

over M+

th___eetins

a division

is a v e c t o ~

ring

h.

to construct

the d i v i s i o n

ring

h

as well

as the vector

space

V. Since ~ ( M t)

~

is primitive,

of all e n d o m o r p h i s m s

each element

r E ~,

it has a faithful of the additive

irreducible

group

M+

the m a p p i n g r :x - xr

(x E M) +

is clearly

an e n d o m o r p h i s m

of

h.

space

M

.

Consider

~ * = Jr* 6 ~ ( M +) I r ~ ~}

the subset

~-module

is o b v i o u s l y

M.

The set

a ring.

For

154

of

£(M+).

The m a p p i n g

~ : r ~ r*

x(r*+s*)

is a h o m o m o r p h i s m

= xr*+xs*

= xr + x s

from

~

onto

~*,

for

= x(r+s)

and x(r*s*)

hold

for every

every

x E M,

x E M+ i.e.

=

(xr*)s*

and

=

(xr*)s

r,s E ~.

x(r - s) = 0.

If

Since

=

xrs

r* = s*, M

then

xr* = xs*

is faithful,

we have

is valid

r = s.

for

Thus

is an isomorphism. Next,

consider

the set

& = [a E ~ ( M +) I ar* = r*a the so-called

centralizer

of

~

in

for every

~ ( M +)

(g

r* 6 ~},

is,

in fact,

the c e n t r a l i z e r

of

~*

I

in

£(Mt)).

identity. quently

We Let

each

claim

that

&

0 ~ d E g. y E M

has

is a division

Then

there

ring.

is an

&

x E M

is o b v i o u s l y such

that

a ring w i t h

xd ~ 0

and conse-

the form

y = xdr* = xr*d =

(xr)d +

for some some

r E ~

z E M.

which

shows

For every

that

element

d

maps

d

maps

isomorphism must be,

of

~

onto

0

M+

onto

M+

of course,

in

onto

itself.

Suppose

zd = 0

for

r* ~ ~, we have

0 = zdr* = zr*d = Hence

M

(zr)d.

and by I.i it follows and c o n s e q u e n t l y

that

d

z = 0.

Thus

has an inverse

d

in

is an

~ ( M +)

which

g. +

These and by

considerations

the i s o m o r p h i s m

1.3 LEMMA. ~(L,V)

such

mensional x~(W)

that

for every

(0) = ~

The proof

irreducible n >

1

that

M

~ : ~ ~ N*, N ~

b e~

V

primitive

and

element

~(W)

is by induction

x~(W)

and assume

an a r b i t r a r y

xa # 0

which

x # 0

that

N

is a subrin$ o f

If

W

is a finite

of

on the d i m e n s i o n

we have

the v a l i d i t y

x~ = V,

n

W

i__nn ~,

of

W.

&

di-

then

If

for o t h e r w i s e

a E ~(W)

of the statement

where

x E V\W.

is evidently

is a n o n z e r o

such an element

ring

V

n = 0,

then

would

not be

Then

V

~-module.

element

that

ring and suppose

is the a n n i h i l a t o r

space.

that

the d i v i s i o n

~(5,M+).

x E V\W.

W = U+[Xn}

such

space over in

irreducible ~ - m o d u l e .

splits into a direct sum Take

is a v e c t o r

can be embedded

is a faithful

and so for every

a faithful Let

V

subspace o f

= V PROOF.

~

Let

show

submodule does not

U

We shall

equivalent of

V

exist,

to

exhibit ~(W)

and so it must i.e.

for

n - i.

is an (n-l)-dimensional

~(W)

an element

# ~(V). equal

= u% ~(V).

vector a E ~(W)

This will V.

Assume

imply

that

By hypothesis

155 Xn~(U)

= V, so for every element

Xna v = v. thus

If

bv

v E V, there is an

is another element with

av - bv E ~ ( U )

N ~ ( x n) = ~ ( W ) .

av E ~ ( U )

Xnbv = v, then

For any

such that

Xn(a v - b v ) = 0

and

x E V\W, define a mapping

~x:V ~ V

by

V,x The element

xa

xa v

(v E

V).

is not n e c e s s a r i l y unique,

v Indeed,

valued. hence

a

=

= xb

v v show next that ~x

nevertheless

the m a p p i n g

~x

is single-

a - b E ~ ( W ) = ~(V) implies xa - Xbv = x(a v - b y ) = 0, and v v v holds for every x E V. C o n s e q u e n t l y ~x is single-valued. We

is

a homomorphism of

the ~-module

V

onto itself. Since

av+ w

is an element w i t h the property x n av+w = v + w

= Xna v + x n a w = Xn(av + aw)'

we have (v+w)~

Since

x =

Xav+ w

=

xa v + x a

w

= V~x+W%x.

Xnavr = vr = Xnavr , we obtain (vr)~x = Xavr = Xavr = (V~x)r'

Hence

~x

is indeed a module h o m o m o r p h i s m for each

= (vr)~ x

shows that

there is a

v E V xa

w h i c h implies y~(U)

= 0

~x

such that =

Xav

x E U+[Xn]

= V~x

=

(Xna)~x

~Xn}

for every

a t ~A(U).

(V~x)r a E ~(U),

Consequently

By the induction hypothesis,

x - Xn~ x E U ~ W.

generated by

Xn, ~x E &

E W, c o n t r a d i c t i n g the hypothesis

Applying

Moreover,

Since for every

(Xn~n)a

=

y E U, and we obtain

1-dimensional subspace

h.

a = a . v

x a = v, we can write n

(x- Xn~x)a = 0

implies

x E V\W.

belongs to the centralizer

that

Since

implies

x n&

is the

Xn% x £ [Xn] .

Hence

x £ V\W.

1.2 and 1.3 we can easily derive Jacobson's Density Theorem for

primitive rings. 1.4 THEOREM. ring ri9~

£(&,V)

Every primitive ring

i__ssisomorphic to a dense subrin$ of the V

over a division

g. PROOF.

By 1.2 the ring

N

can be embedded into a ring

is an irreducible faithful ~ - m o d u l e Take

~

of all linear transformations on a vector space

n

linearly independent vectors

y l , . . . , y n E V. xlr = Yl

If

n = i, then by

and hence

there is an element differ from

Yn"

~

such that

Xn~ = V.

Xl,...,x n ~ V

Assume

~

xir = Yi

such that in

V

£(V)°

r E ~

such that

is (n-l)-fold transitive. for

i = l,...,n-l, but

~ = ~(Xl,...,Xn).

r + s E £(V)

~

and arbitrary vectors

Hence there exists an element

The linear t r a n s f o r m a t i o n

£(g,V)

is the centralizer of

Xl~ = V, there exists an

is transitive.

r E ~

~

C o n s i d e r the a n n i h i l a t o r

cable and we obtain XnS = Yn"

and

Now, s E ~

Xnr

Then may

1.3 is applisuch that

is now the desired one, namely

156

xi(r + s )

= Yi

holds

ring of

&(V)

for every

for each

i = l,...,n.

n >

i

which means

Thus

~

is an n-fold

transitive

that

~

is dense in

~(V)

It is also true that every dense subring of not use this statement. the usual one theorem,

(cf. Divinsky

consult

Jacobson 2.

[i] and Kert~sz

([7], Chapter

cerning

and semisimple

inverse

complete direct is linearly

rings.

of the density

for the study of linearly compact

We shall prove some elementary

sum of linearly

compact modules,

compact modules

statements

con-

further we shall prove that the

endowed with

the product

topology,

compact.

A

topolo$ies

be a directed

2.1 DEFINITION. Mc~

For a discussion

but we shall

a version of

II).

the preparations

limits of linearly

Only Hausdorff Let

[i]).

is primitive,

1.3 is actually

INVERSE LIMITS AND LINEARLY COMPACT MODULES

In this section we continue primitive

£(~,V)

The proof of 1.4 along with

sub-

and continuous

will be considered.

set of indices.

A system

~ = iMc~,~ ~ I c~> ~ E A}

homomorphisms

~ :c~M c ~ ~ M~

of topological lE-modules

is said to be an inverse

system if

m~ = c ~ holds for every ~ > ~ > ~? and TT~ is the identity mapping on M P? every c~ C A. The inverse limit M = li~m [~ of the inverse system ~ is the topological

submodule

(...,x~, . . . , x , ...) E topology of

M

M

~l M c~A (2

such that

is the product

2.2 PROPOSITION. IIM . c~A ~ PROOF.

of the complete

sum

[~ M consisting all vectors ~ A (2 for every ~ > ~ E A and the

x~c~~ = x

topology

The inverse

direct

limit

We shall show that the set

relativized

to

M = li~m f]

[ M \M

for

M.

is a closed submodule

is open.

of

Take an arbitrary

~A element

(...,x

ponents

K

II M \M. There exist indices cY> ~ such that xc~ ~ # x~. c~A c~ Let K = (...,K~,...,Kc~,...) be an open subset of ~ M such that K~ and K eg:A :~ c~ are open sets with x~ E K~, Xc~ E Kc~ and K ~c~ ~ K~ = ~ and all the other comequal

are Hausdorff clear that Let

,...) E

K M

.

Observe

spaces

does not contain elements be a linearly

M

that the topological

and hence also regular

for the neighborhood ~-modules

M

= M/K

so that such

of

M.

topologized ~ - m o d u l e

system of

0

consisting

form an inverse

system

modules

Hence and

K~ M

does exist.

It is

is closed.

K = [K~o~A

of submodules. ~

under consideration

together with

be an open base

Obviously

the

the homomorphisms

157

M/K ~:M/K

--

if/ K~/K

2 . 3 PROPOSITION ( g e l i n s k y M

i_~s c o m p l e t e , PROOF.

then

M

Taking

~ ~ A, i.e.

x = 0

and

the mapping M

~,

M

forms

~p = ( . . . , x

~ ~ A.

into

li~ ~.

Moreover,

if

t__oo l i ~ . ,...)

with

Further, mp = 0

x E

M

x E M, x

implies

x

=

6 M , 0

for

is Hausdorff, we must have

is complete and consider an arbitrary element

x

6 M

a Cauchy net

For this element

onto

can be embedded

li~ ~.

y = ( .... y , . . . ) 6 Choose elements

~>

an__~d h o m e o m o r p h i c

where

into

for all

Q K . Since the topology of a£A ~ is an isomorphism.

~

Suppose

M

i__ss i s o m o r p h i c

we have a h o m o m o r p h i s m from each

[1]).

~ M/K~

such that

which x

converges

we have

o

~¢-..lim ~.

xff+Kff = yff

to

xo + K

~

for each

some e l e m e n t = y

~ 6 A,

x° 6 M

for each

~ E A.

since

Then

Ix

M

complete.

Hence

is ~

I ff E A]

maps

M

lim ~. Clearly

~

the

topology

is

in

l
Thus A

is

continuous.

discrete ~

and

is

topological ~ - m o d u l e s

if

~ > ~,

the

K

image

E K. Kff~

Since of

in

K~

each is

M

open

a homeomorphism.

be a set and

d i r e c t e d by inclusion.

Take an open submodule K~ ~_ Kff

F

be the set of all finite subsets of

C o n s i d e r the finite direct sums endowed w i t h the product topology.

A.

Then

F

is

n M (f E F) of ~6f ~ If g c f E F, then f

M can be mapped n a t u r a l l y onto ~ M by a p r o j e c t i o n ~ . Now o~f ~ c~g ~ g = [ ~ M ^ , ~ f I g c f E F] forms an inverse system of finite direct sums. ~f ~ ~ -2.4 PROPOSITION. li_m ~ i_!s isomorphic a n d homeomorphic to the complete direct sum

n M ~A ~ PROOF.

endowed with the product topology. Define a m a p p i n g

( .... x .... where

x f = x 1 + . . . +Xffn

for

~: ~ M ~ lim ~ c~A ~ -

by

)~ = (...,xf,...) the

indices

ffl,..,,ffn

E f.

Obviously

~

maps

isomorphically onto li~ ~ and is continuous. Taking into account that the M c~ oEA product topology is the w e a k e s t one such that p r o j e c t i n g onto the components we get back the topology of the components,

it is obvious

that

~

is a homeomorphism.

We continue this section with some elementary properties of linearly compact modules.

The following assertion, w h i c h will be useful later, provides

tunity for another d e f i n i t i o n of linearly compact modules.

the oppor-

158

2.6 PROPOSITION if it is linearly

(*)

~

property,

(Leptin

topologized

PROOF.

Conversely, A

+ K k.

M

_G = {a

and

N

2.7

intersection

N G =

linearly

M.)

O (a

If

N

compact

Let

K = {Ks}c~ A

such that each

s

the intersection

K

a

E (a+Ks)

Q N.

in

(a + K Clearly

a E N

s + (K

PROOF.

= {a S + K S } o ~ A

Ovbiously

G

it

topolo$ized then

N

is

the neighborhood

Take an element

~-module closed

system

a E N.

M

in of

M. 0

in

Now for each

Take also an element

~ N) c ( a + K )

Q N.

and so

For any element

ba E a + (K S N N).

Hence also

Let

~

A} = i ( a + K s )

of

N

0 N Is E A}

with the finite intersection NF

property.

has an element in

O ((a+Ks) NN) c_ q ( a + K ~ ) c~A c~A

N.

Hence

= a

N = N. be a continuous

homomorphism N.

Then

of a linearly compact Mqo

i__sslinearly

compact

topology. topology of

M~

is clearly linear.

is any family of closed subvarieties then

is linearly compact, a nonempty

M.

for

the intersection

~ N)) =

The relativized

section property,

M.

(*), we have

a linearly

into a linearly topolo$ized ~-module

in the relativized

is an open base

of

Moreover,

and thus

2.8 PROPOSITION. M

property.

is always open whenever

topology,

is nonempty.

b s - a~ E Ka N N

is linearly compact, ~aN (as+ ( K

of

relativized

be a n o p e n b a s e

*s a family of open subvarieties

~-module

family

+ K x ) = Q ( a a + Z o) = N F

f = { a s + ( K S N N) I ~

that

a submodule

a submodule

the

) ~ N

) Q N ~ a + (K Q N). -- s s

implies

linearly

and it consists of open subvsrieties

is a submodule of

s

bs E (a +Ke) N N, we have

N

~}~CA

Hence by condition

+Z

is

linearly

M

Since

Take an arbitrary

having the finite intersection

consisting of submodules

property

is closed,

(*).

compact.

PROPOSITION.

PROOF.

(a + K

0

(As it is easy to verify,

is

in any linear topology

+ Z ~ + K k Is E A, % ~ Al, where

system of

an open submodule of

is

which has the finite intersection

suppose that (*) is fulfilled.

0 #

is linearly compact if and only

the following condition:

the validity of condition

for the neighborhood has the finite

M

intersection.

of closed subvarieties

Form the family

and

and satisfies

Since every open submodule implies

f = {a +Z~

a +Z

An ~-module

family of open subvarieties

ha___~snonempty

compactness

contains

[i]).

K -i = ~ (a S

K -i

intersection.

+ K)~- i}c~A

of

M~

If

with the finite inter-

is such a family in

has a nonempty intersection.

Consequently

M. also

Since K

M has

159

2.9 PROPOSITION.

If

topologized ~ - m o d u l e PROOF. homomorphism

M/A

M,

By 2.7, A

A

and

then and

~:M ~ M/A

M

is closed in

is closed.

Hence

differ from k:a~+K~

it is sufficient

varieties of Let

has the form

M~.

Project

~ a ~+K

a

M

).

. c~

E [~F k-~

~

=

ak+K k

~ K a~A c&

Bq0~ I = A + B

Since

M

c~

Let

F_ = [ a % + K k } x E A

M.

-~F = ~a ~ + K

is linearly compact, N F k-c~

and take the element

a

= (...,a o

ak+ ~AK

K %

for each ~EA

of open sub-

is nonempty for each

,...) ~

II M . ¢~A ~

~

Then

a

~.

is o

since only a finite number of K k 's differ

k

and F has the finite intersection property. ~y ~I M is linearly compact. c~A ~ 3.

be such a

where only a finite number of

M

that

is closed in

to consider only families of open

onto the ~-th component

Thus we have a family

obviously contained in each from

is linearly compact in

~ M of linearly compact ~A ~ in the product topology.

_is _ _linearly _ compact

K%

and so the natural

The complete direct sum

In view of 2.6,

Each

~

In view of 2.8, Bq~

subvarieties with finite i n t e r s e c t i o n property. family.

M.

are closed submodules of

is continuous.

2.10 PROPOSITION.

PROOF.

are linearly compact submodules of a linearly

A + B B

and thus again by 2.7, Bop

S-modules

B

Hence

a

~ ~F -

o

proving

LINEARLY C O M P A C T PRIMITIVE RINGS

We now turn to structure theorems for linearly compact primitive rings. 3.1 PROPOSITION. topolosY

continuous image o f ideal of

~

is a primitive rin$ endowed with a linearl~ compact

~.

Moreover

M ~ ~ ~/K, where

K

M~

which is a

is an open maximal right

2.

PROOF.

Since

For any element e E ~

If

T, then there exists an irreducible faithful S - m o d u l e

~

is primitive,

0 ~ m ~ M, we have

such that

me = m.

there is an irreducible faithful ~ - m o d u l e mO~ = M

Consider

M.

and hence there exists an element

the set

J = [y = x - ex Ix ~ ~ }. Then J

J

is obviously a right ideal of

h o m o m o r p h i c a l l y and continuously,

mapping.

As it is easy to verify,

relativized topology.

Hence

Take an open right ideal

maximal excluding

e.

of

~

is Hausdorff,

of

~,

for if

that

K~

J

~.

and

The m a p p i n g ~

x~ = x - ex

relativized to

the topology induced by

J ~

maps

~

is the identical on

J

is just the

is linearly compact and, by 2.7, closed in K

of

~

such that

e ~ K

By Zorn's lemma such right ideal even regular[).

This right ideal

is a right ideal properly containing

and K

K

onto

J c K

does exist

and

~. K

is

(the topology

is s maximal right ideal K, then

e E K~

implying

160

r = (r- er) + e r for every

r ~ ~

and

Consider

~/K

image of

is clearly

topology

system of

~(M)

0

of

K

m E M = ~/K.

We can take

isopen~ s o i s J(m).

a continuous

compact

PROOF.

If

~.

Applying topology ~(V).

2.7, ~

in

is closed

REMARK. weakest

one

of

~(V),

is a linearly of

~

£(V).

and

Let

direct

~(V)

so

x E J(m)

~

onto

compact

means

£(V)

primitive

rx E K.

that

V ~ ~/K

ring,

for some vector

the finite

Since

then there is

space

V

Consequently

V

over a

topology. ~

is a dense subring of

is a vector

£(V)

space over the centralizer topology,

compact

primitive

ring

~ : ~ ~ ~(V)

is also a homeomorphism.

be a vector

space over a division

~

ring

is a

&.

sum

Then

~I J of minimal idempotent £ ( V ) - i s o m o r p h i c aEA ~ topology 7 of I] J induces the finite topology

is a weakest

Take a basis

{x

linearly

I ~ ~ A}

compact

of

V

(Hausdorff)

and consider

topology of

right ~(V) ~(V).

the right annihilators

~J((~l..... (Yn) = {a ~ ~(V) I x(~ a = 0, i = i ..... n} i and

J(~l ..... ~n ) = {a C ~(V) I x~a = O, ~ E A\{~ I ..... ~n ]} for every

finite subset

J(~l,...,~n) J~i

=

J(~i).

By 11.1.5,

{~l,...,~n}

is an ~ - m o d u l e Further,

each

of

A.

and a direct

sum

J~i

It is easy to verify J(~l,...,~n)

is an idempotent

each

J~i

is generated by an idempotent

If the image of

x i

under

e i E ~(V)

by

~ = ~ = £(V).

If the topology of the linearly

The product

PROOF.

the open base

is linearly compact also in this weaker

then the mapping

J .

the topology with

~.

~(V), where

Since

in

is a complete

ideals

m = r+K,

is endowed with

3.4 PROPOSITION. ~(V)

i.e.

of the right ideals

0]

1.4 and 1.2, we obtain

in the finite ~

~i ~

~(V)

A

of

topology

isomorphism

ring

~/K

and

N J(m) = 0 since M is faithful. We have proved mEM The finite topology ~(M) o f ~ is Hsusdorff and weaker

3.3 PROPOSITION.

division

irreducible ~ - m o d u l e

Moreover,

3.2 PROPOSITION. than the linearly

~,

consisting

J(m) = [x~ ~ I m x = for every

faithful

a

~.

the finite

for the neighborhood

~ = K~

K ~ = ~.

The factor module is s continuous

~ L+K

minimal

is given by

e~i,

that

=

~ J of ~-modules i=l ~i right ideal of £(V).

i.e.

d~i = e~i~(V).

161

e xai ~i then

e

2

= e

ai

~

j=l

pjx

implies ai

2 x ie i

=

0 # Xaieai ~j = a i

Hence

i = l,...,n. Now

Consider

a linear = x

x~ke~kae~i q~:e

r ~ e

ak

ae

ak

~ ,

.

J-J-l~jx{jeai

r

xai e ~i = o.x i ~i

transformation

# 0

which

a 6 ~(V)

shows

that

e

~i

is an ~(V)-homomorphism

between

right

~(V)e

ideals

for

xaka

i =

'

X

PkPi a i The mapping

# O. ai

J

and

J

ak Since

(Pi @ 0)

such that

~k

=i

the zero mapping. minimal

=

for one of the ~j 's involving

have

we

(9j ~ &),

~j

, and

~

is not

ai

J

and j are irreducible ~(V)-modules (as they are ak ai ~6(V)), ~ is an isomorphism. Hence all the J 's are iso-

of

morphic £(V)-modules. We prove next

that

£(V)

is a direct

sum of

J(a I .... ,an) n ~(al, ...,an) = O.

Clearly a E ~(V)

a linear

and choose

x~b = We now have

transformation X a. I

if

0

otherwise

b E J(al,...,~ n)

and

J(al,...,an)

Take an arbitrary b E ~(V)

and

linear

~(~l,...,an). transformation

such that

~ ~ [~I ..... ~n }

s - b E ~I(~l,...,a n)

implying

a = b + (a- b) E J(~l .... '~n ) +~(al'''''an)" Thus

~(V)

is the required

direct

J(~l,...,~n) Consider

the inverse ~ = [

~

the direct

J

, ~f I f,g

are finite

~ aiEf

are endowed with

J

topology

~(V)

for neighborhood of

compact

the product

topology

(which

is of

£(V).

finite

system of

By III.9.2,

~(V)

0

subset in

[a I ..... a n} ~ A}

£(V),

is complete

we obtain in

~(V) o

the finite Hence

2.3 and

and we obtain ~(V)

linearly

A},

Choosing

2.4 are applicable

Now 3.2 implies

of

~i

= [~(a I .... ,an) I for every for an open base

subsets

g

sums

course discrete).

~ £(V)/~(a I ..... ~n ).

system

ai~f ai where

sum, and in addition

~

that

~(V)

in

~(V).

l~im Q ~

~ J . ~A ~

is s weakest

topology

of

£(V)

and by 2.10, ~(V)

is

.

162

C o m b i n i n g 3.3 and 3.4, we arrive at 3.5 THEOREM. i)

For a rin$

~,

the followin$ statements are equivalent.

~

is a primitive ring endowed with s weakest

~

i__ssisomorphic a n d h o m e o m o r p h i c

linearly compact

(Hsusdorff)

topology. ii)

formations on a vector space with the finite topology iii)

~

~-isomorphic

V

to the rin$

£(•,V)

over a d i v i s i o n rin$

of all linear trans-

A, where

~(&,V)

is endowed

~(V).

i_~s isomorphic and homeomorphic

to a complete direct sum

idempotent minimal risht ideals

product topology and each

J

J , where

~ J of ~6A ~ - is endowed with the

n J

is discrete.

It is now clear that a linearly compact primitive ring always has a nonzero socle, A topological ring

~

will be called topolo$icall~ simple,

its only closed ideals and 3.6 PROPOSITION. PROOF.

1.4.3.

~(V),

to show that any ring

is topologically simple.

Ideals of

It follows from 1.2.10 that every nonzero ideal of

transitive for

0

and

~

are

A linearly compact primitive rin$ i__~st o p o l o g i c a l l y simple.

In view of 3.5, it suffices

the finite topology

if

~ 2 # 0, cf. 1.3.5.

n = 1,2,...,

and so by III.9.1,

~(V),

endowed with

~(V)

were found in

~(V)

is dense in

£(V)

is n-fold in the finite

topology. 4.

L I N E A R L Y COMPACT S E M I S I M P L E RINGS

Before dealing with linearly compact semisimple rings, we will prove some preliminary propositions. if

~/P

primitive ideal of

p

P

of a ring

Hence

~

If

P

is a primitive

is a maximal closed ideal of

PROOF.

~

We will prove that

such that

p

of 3.1, one csn show that

P

~.

M

Taking into account that m E M

0 ~ mU.

w h i c h shows that

implies that

P

= ~

M

~/P M

with

N,

in

is primitive,

can be considered as ~.

As in the proof in the discrete

holds with a suitable open m a x i m ~ l right ideal

that there is an element UnP

Since Then

is a linearly topologized ~ - m o d u l e

3.

Hence

M.

is just the annihilator of

M ~ ~/K

a E ~\P.

ideal, is a

ideal of a linearly compact rin$

is closed in

topology and that Let

0

~.

there exists sn irreducible faithful ~ / P - m o d u l e an ~ - m o d u l e

is called a primitive

is primitive if and only if

N.

4.1 PROPOSITION. then

An ideal

is a primitive ring.

M

ms # 0 P

is a maximal closed ideal of

is a faithful ~ / P - m o d u l e , and a n e i g h b o r h o o d is closed in ~.

~.

U

of

Finally,

K

of

we deduce a 3.6

with

163

A set

~

consisting of primitive ideals

P , ~ E A, of a linearly compact ring

is said to be an idependent system if for every finite subset {P I,...,P } n n n of ~, we have ~/ n P -~ N ~/P . Using Zorn's lemma, it follows that every i=l ~i i=l ~i linearly compact ring ~ has a maximal independent system e 0 of primitive ideals. 4.2 PROPOSITION.

Let

e I = {PI~}c~A

and

@2 = ~P2~}~EB

be two maximal in-

dependent systems of primitive ideals of a lines!l y compaqt rin $ weakest linearly compact topology. PROOF.

N

c~API~

=

N P

~

Suppose that the statement is not ture.

we may assume that

~API~N ~-- ~B~ P2~"

such thatn o~API~n ~_ D 2. D1 =

Then

N P ~ D2° i=l i~ i --

D2

endowed with a

Without loss of generality,

There is a maximal closed ideal

Hence for any finite subset

Since

2~

~

.

i~l,. .°,~n}

of

D 2 = P2~ o E @ 2 A, we hsve

is a maximal closed ideal, by 2.9 we have

D 2 # D I + D 2 = D I + D 2 = 2. Thus

~/D 1 N

D2

splits in a direct sum of

Since the topology of

~

DI/DI[~ D 2 ~ ~/D 2

and

D2/D 1 N D 2 ~ 2/D I.

is a weakest one and the isomorphism

~:~/D I N D 2 ~ ~/D I ~ ~/D 2 is continuous, ~

is a homeomorphism.

system of primitive ideals of

containing

lel,D2}

is an independent

~, contradicting the msximality of

Since for any primitive ideal @

Consequently

p

of

~

@i"

there is a maximal independent system

P, 4.2 immediately implies

4.3 PROPOSITION.

Let

e = ~P

I ~ E A}

primitive ideals of a linearly compact ring

be a maximal independent system of 2.

Then

N p

e~gals the intersection

o6A ~ of all Primitive ideals of 4.4 DEFINITION. primitive ideals of

~.

A ring ~

equals

~

is semisimple if the intersection

>(~)

of all

0.

In a terminology consistent with 11.5.4, a semisimple ring should be called semiprimitive. only if

~

In view of II.5.4, it is clear that a ring is a subdirect sum of primitive rings.

primitive ideals of

~

2

is semisimple if and

The intersection

~(~)

is referred as to the Jacobson radical of the ring

of all ~.

For

more information on this subject, consult Jacobson ([7], Chapter I). Linearly compact semisimple rings have several interesting characterizations. Some of them are presented in the following result.

Characterizations

are due to Leptin [2], whereas iv) can be found in Wiegandt 4.5 THEOREM.

Th___eefollowin$ conditions on a rin$

~

ii) and iii)

[i].

are equivalent.

164 i)

~

compact

is a linearly compact semisimple ring endowed with a weakest

(Hausdorff)

ii)

~ ~

~ £(f~c,V ), algebraically and topologically,

the right hand side is the product ,~.(V ) , (y

C~E

iii)

where the topology of

topology of the finite topologies

~ J , where each

J

v~c ~

~

is an idempotent minimal

right ideal, J

is s linearly compact r e g u l a r ring endowed with _a weakest

(Hausdorff)

PROOF.

topology

i) = ii).

primitive ideals of [ a I .... ,C~n}

of

the

~.

is discrete. linearly

(regularity is m e a n t in the algebraic sense of 1.3.5).

Choose a maximal

independent

By 4.3, we have

A, we have n

system

~ P = 0 a6A ~

~ = iP(~ I ~ 6 A}

of

and for any finite subset

n

~/ ~ p

~

i=l a'i

~ '~/p

i=l

~i

These finite direct sums obviously form an inverse system 111.5.9,

of

v

topology of the right hand side is the product topology and each

compact

~(V )

A.

~ ~

iv)

linearly

topology.

[~.

Taking into account

and 2.3 and 2.4 of this appendix, we deduce

In view of 3.5, ~ / p is weakest,

~--~ lira [~ ~

N/P tEA

~ £(&

holds for each

,V )

the isomorphism

~

11 ~ / P

c~ E A•

Since the topology of

is also a homeomorphism.

a6A ii) = iii).

This follows directly from 3.5.

iii) = iv). that every

J

= e ~

By 11.1.14, and

&

a E J , there is an

(evre?)-i = eyse? a(e

Thus for each

s)a

in =

~ E C, J

~ E C, there is an idempotent

is a division ring with

r E ~

A v. (e

for any

= e ~e

such that

Consequently,

r)(e

s)(e

r)

=

(e

is a regular ring•

~

is also,

Note that the discrete

E J e .

such Hence for

and

e re VV

a E JV

and

e?s E Jr' we have

)(e

se

)r

Thus

has an inverse

= e r = a.

~ J

~c hence

e

a = e r V

for re

the identity

are regular rings and v

topology in each

is a w e a k e s t one,

J V

so the product topology yields again a weakest one. iv) = i).

Consider an arbitrary element

existence of an element

r E ~

such that

z E ~(~).

z = r - zr•

We shall establish

the

To this end, consider

the

right ideal J = j r - zr I r E ~ } . According

to II.7.6, J

is a m o d u l a r right ideal of

lemma insures the existence of a m o d u l a r right ideal

~. M

Suppose

z @ J.

Then Zorn's

which is maximal with

165 respect to the exclusion of ideal

N

z.

properly containing

In fact, M M

r = (r - zr) + z r ~ M + N holds for every

r E ~.

is a maximal right ideal,

contains

Obviously

M

z

for any right

and thus

~ N

is modular.

Consider next the two-sided

ideal P = [y E ~ l~y ~ M}. Since

~

is regular,

it follows that

~/P-module.

there is an

y = yxy E M

Consequently

x E ~

implying

p

with p c M.

yxy = y.

Hence for every

Moreover, ~ / M

is a primitive ideal of

~.

y E P,

is a faithful

By hypothesis, we have

z 6 ~(~) c P c M c o n t r a d i c t i n g the assumption element

r 6 ~

such that

z ~ M.

Consequently

z E J

Again take an arbitrary element

a E ~(~).

x E ~

such that

axa = a.

element

r ~ ~

such that

xa = r - xar, and we obtain

a = axa = ar - a x a r ~(~) = 0, i.e. 4.6 COROLLARY.

~

Then

Since

element

Thus

4.7 C O R O L L A R Y

is semisimple.

A semisimple ring

~ N

endowed with a weakest linearly compact i__~stopologically simple.

(Leptin

[2]).

~ linearly compact ring

of

A ring

is finite for all

~.

~

is topologically simple

Trivial by 4.5 and 4.6. ~

is linearly compact,

a__n_nopen base for the n e i g h b o r h o o d system of

PROOF.

there is an

there is an

N ~ £(~,V).

4.8 COROLLARY.

dim V

By the above,

S t r a i g h t f o r w a r d by 3.6 and 4.5.

if and only i f PROOF.

is regular,

xa E ~(N).

ar - a r = O.

=

topology i~s primitive if and only if PROOF.

and thus there exists an

z = r - zr.

Necessity.

Without

0

semisimple and its ideals f o r m

if and only if

~ ~

n £(&

,V )

and

~ E A. By 4.5, we have

R £ ( V ). ~A loss of generality, we may assume that ~/U ~

~ ~

C o n s i d e r an open ideal

U

~ £ ( V ). finite

Since

H £ ( V ) is discrete, each component ~ ( V ) is discrete in the finite finite ~ n topology. H e n c e 3.5 implies that each ~ ( V ) splits into a finite product ~ J. i= 1 1

of minimal idempotent ~ - i s o m o r p h i c

right ideals

J..

In this case the dimension of

i

each

V

must be finite,

for

dim V

= n .

S u f f i c i e n c y is obvious.

166 4.9 COROLLARY.

A rin$

topology if and only if PROOF. £(V )

~

~ ~

For necessity,

must be finite since

4.10 COROLLARY.

PROOF.

Trivial

Finally,

~

the terms '~primitive"

(cf. Leptin compact.

system

dim V

is finite.

imply

0

~

i__sssimple if and only if

rings are always prime and, by II.i.25, is valid,

and "semisimple" respectively.

in

~, then

Further

can no longer be replaced by the terms For let

~

denote

is considered p

for rings with

for linearly compact rings in 3.5 and 4.5

the ring of all p-adic

as an open base for a

induces a linearly compact

~i = p~' as s closed submodule of

topology on

~, is linearly

(Prove that a closed submodule of a linearly compact S-module

linearly compact in the relativized exactly the ~l-SUbmodules,

~i

to see, ~i

has no primitive

two nonzero

ideals

and thus

where

~ = {pi~ i i = 1,2,...j

[i]).

is again trivial.

is finite.

"prime" and "semiprime",

neighborhood

JK # 0

linearly compact,

~.

and the number of the components Sufficiency

A linearly compact tins

dim V

Even though primitive

If

is finite for all

i__ssprimitive and linearly compact in the discrete

~ ~ ~(~,V)

nonzero socle also the converse

integers.

dim V

is discrete.

4.6 and 3.5 immediately

where

and l inearl~ compact in the discrete

in view of 3.5.

4.11 COROLLARY. ~ £(&,V)

~

A tins

topolosy if and only if

is semisimple

~ ~(V ) and finite ~ 4.8 is applicable,

J, K

topology.)

Since the ~-submodules

is linearly compact as a ~l-mOdule. ideals

of

(cf. Wiegsndt

~i' we have

which shows that

0

[3]).

J = p~,

of

~i

for some ~i"

In addition,

are

As it is easy

On the other hand,

K = pm~

is a prime ideal of

prime, but is not semisimple.

is always

for any

n,m > i,

Hence

~i

the socle of

is ~

equals

0. For further discussion of semisimple [2], Wiegandt

[2] and Eckstein

An obvious generalization compactness.

of linear compactness

Since every semisimple

socle (see Wiegandt

linearly compact rings, we refer to Leptin

[i]. is the notion of local linear

locally linearly compact ring has a nonzero

[4]), the study of their structure

is not the case in the theory of locally compact rings Linearly

compact nonsemisimple

[3] and Arnautov = ~(~)).

[i], and particular

[i], Fuchs

to Andrunakievi@,

Arnautov

(cf. Skornjakov

rings were investigated

by Leptin

[1],[2]). [2], Wiegandt

attention was paid to radical rings

Further results on linear compactness

Fischer and Gross

can be easily handled, which

[i], and Warner

were obtained by Wolfson

[i].

and Ursu [i], Warner

(where

For recent developments

[2], widiger

[i], we refer

[i] and Wiegandt

[5].

167

CURRENT ACTIVITY From the material

of these Lectures,

we may easily extract the following

categories: i.

Objects:

pairs of dual vector spaces,

Morphisms: 2.

semilinear

Objects: weakly Morphisms:

3.

Objects:

complete

lattices

(U-closed

Objects:

rings with a nonzero socle,

ring isomorphisms;

Morphisms: 5.

adjoint;

topologized vector spaces,

maximal primitive

Objects:

with a surjective

iseomorphisms;

Morphisms: 4.

isomorphisms

satisfying

subspaces of

the double covering condition

V),

lattice isomorphisms;

multiplicative

semigroups

of objects

in 3. above

(characterized

abstractly), Morphisms:

semigroup

isomorphisms.

Further categories may be obtained by taking certain subobjects Under some mild restrictions categories

are either equivalent

relationship

or isomorphic.

semigroups,

by the author entitled rings and lattices".

toward a better understanding

of homomorphisms

transformations, pierce

[i].

see Fajans

Multiplicative

[4], Satyanarayana peinado Eckstein

[I]. [2].

widiger

These Lectures

thus represent a step

among these different branches

of linear transformations

and antiisomorphisms [1],[2],

includes mainly in-

of various

semigroups

Jodeit and Lam [i], Mihalev

semigroups

Ill, a comprehensive

of linear

and Satalova

[i],

of certain rings have been studied by Petrich survey of this subject can be found in

Semigroup methods have been successfully

used in ring theory by

Further classes of rings with unique addition have been recently

found by Martindsle investigated

of vector and projective

geometry.

The recent work on semigroups vestigations

study of the

forms the subject of an un-

"Categories

of the relationship

of modern algebra an~ projective

in 3., 4., and 5.

most of the resulting

A comprehensive

among these and many other categories

published manuscript spaces,

on objects of these categories,

[i] and Stephenson

by Andrunaklevlc,

[i], Wiegandt

[5].

Arnautov

]i].

Linear compactness

and Ursu [i], Arnautov

for rings has been [i], Warner

[1],[2],

168

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Jodeit, M. and Lam, T.Y. [i] Multiplicative Johnson,

to the theory of centralizers,

20(1969),10-16.

Proc. London Math. Soc. 14

R.E.

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Archiv Math.

B.E.

[l] An introduction (1964),299-320. Johnson,

maps of matrix semigroups,

R.E. and Kiokemeister,

Proc. Amer. Math.

Soe. 9(1958),57-61.

F.

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Trans.

171

Kaplansky, I. [i] Topological rings, Amer. J. Math. 69(1947),153-183. [2] Locally compact rings, ~ner. J. Math., Part I, 70(1948),477-459; 73(1951),20-24; Part III, 74(1952),929-935.

Part II,

[3] Ring isomorphisms of Banach algebras, Canad. J. Math. 6(1954),374-381. [4] Fields and rings, Univ. of Chicago Press, Chicago and London, 1969. Kelley, J.L. [i] General topology, Van Nostrsnd, New York, 1955. Kertesz, A. [i] Vorlesungen (iber artinsehe Ringe, Aksd~miai Kiad$, Budapest, 1968. K@the, G. [i] Topological vector spaces I, Springer, New York, 1969. Lambek, J. [i] Lectures on rings and modules, Blsisdell, Waltham,

1966.

Lefschetz, S. [i] Algebraic topology, Amer. Math. Soc. Coil. Publ. Vol. 27, New York, 1942. Leptin, H. [i] Uber eine Klasse linear kompskter abelscher Gruppen, Abh. Msth. Sem. Hamburg, Part I, 19(1954),23-40; Part II, 19(1954),221-243. [2] Linear kompakte Moduln und Ringe, Math. Zeitschr., Part I, 62(1955),241-267; Part If, 66(1957),289-327. Msckey, G.W. [I] Isomorphisms of normed linear spsces, Ann. Math. 43(1942),244-266. [2] On infinite-dimensionsl 155-207.

linear spaces, Trans. Amer. Math. Soc. 37(1945),

MacLane, S. [i] Extensions and obstructions

for rings, Illinois J. Msth. 2(1958),316-345.

Martindale, W.S. [i] When are multiplicative mappings additive? 695-698.

Proc. Amer. Math. Soc. 21(1969),

McCoy, N . H [i] The theory of rings, McMillan, New York, 1970 (7th printing). Mihalev, A.V. [i] Isomorphisms of endomorphism semigroups of modules, Algebrs i Logika, Part I, 5, no. 5 (1966),59-67; Part If, 6, no. 2 (1967),35-47 (in Russian). Mihalev, A.V. and Satalova, M.A. [I] Automorphisms and sntiautomorphisms of the semlgroup of invertible matrices with nonnegative elements, Mat. Sbornik 81(1970),600-609 (in Russian); Transl. Math. USSR-Sbornik 10(1970),547-555.

172

Morita, K. [I] Category-isomorphisms and endomorphism rings of modules, Trans. Amer. Math. Soc. 103(1962),451-469. Muller, U. and Petrich, M. [i] Erweiterungen eines Ringes durch eine direkte Summe zyklischer Ringe, J. Reine Angew. Math. 248(1971),47-74. [2] Translationshulle and wesentlichen Erweiterungen eines zyklishen Ringes, J. Reine Angew. Math 249(1971),34-52. Neumann, J. von [i] On regular rings, Proc. Nat. Acad. Sci. 22(1936),707-713. Ornstein, D. [I] Dual vector spaces, Ann. Math. 69(1959),520-534. Peinado, R.E. [i] On semigroups admitting ring structure, Semigroup Forum i(1970),189-208. Petrich, M. [i] The translational hull of a completely 0-simple semigroup, Glasgow Math. J. 9(1968),i-11. [2] The semigroup of endomorphisms of a linear manifold, Duke Math. J. 36(1969), 145-152. 13] The translational hull in semigroups and rings, Semigroup Forum 1(1970), 283-360. [4] Dense extensions of completely O-simple semigroups, J° Reine Angew. Math., part I, 258(1973),i03-125; Part II, 259(1973),109-131. [5] Introduction to semigroups, Merrill, Columbus,

1973.

Pierce, S. [i] Multiplicative maps of matrix semigroups over Dedekind rings, Archly Math. 24 (1973),25-29. Pless, V. [i] The continuous transformation ring of biorthogonal bases spaces, Duke Math. J. 25(1958),365-371. Plotkin, B.I. [i] Automorphism sroups of alsebraic system~, Nauka, Moscow,

1966 (in Russian).

R~dei, L. [i] Alsebra, Vol. i, Pergamon, Oxford and New York, 1967. R@dei, L. and Steinfeld, O. [I] 0her Ringe mit gemeinsamer multiplikativer Halbgruppe, Comment. Math. Helv. 26(1952),146-151. Ribenboim, p. [i] Rings and modules, Interscience, New York, 1969.

173 Rickart, C.E. [I] One-to-one mappings of rings and lattices, Bull. Amer. Math. Soc. 54(1948), 758-764. [2] Isomorphic groups of linear transformations, Amer. J. Math., Part I, 72(1950), 451-464; Part II, 73(1951),697-716. [3] Isomorphisms of infinite-dimensional analogues of the classical groups, Bull. Amer. Math. Soc. 57(1951),435-448. Rjabuhin, Ju~ M. [i] Rings with unique multiplication, 77-78 (in Russian).

Izv. Akad. Nauk Mold. SSR, No. i (1963),

Rosenberg, A. [i] Subrings of simple rings with minimal ideals, Trans. Amer. Math. Soe. 73 (1952) ,115-138. Rosenberg, A. and zelinsky, D. [i] Galois theory of continuous transformation rings, Trans. Amer. Math. Soc. 79 (1955),429-452. Satyanarayana, M. [I] On semigroups admitting ring structure, Semigroup Forum, Part I, 3(1971), 43-50; part If, 6(1973),189-197. Skornjakov, L.A. [I] Einfache lokal bikompakte Ringe, Math. Zeitschr. 87(1965),241-251. [2] Locally bicompact biregular rings, Mat. Sbornik 69(1966),663 (in Russian). Steinfeld, O. [i] On Litoff's theorem, Ann. Univ. Sci~ Budapest 13(1970),101-102. Stephenson, W. [i] Unique addition rings, Canad. J. Math. 21(1969),1455-1461. Sz~sz, F. [i] Uber Ringe mit Minimalbedingung fur Hauptrechtsideale. 7(1960),54-64.

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174

Wiegandt,

R.

[i] Uber linear

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Bull. Acad.

linear kompakte Ringe,

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Poi. Sci.

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Studia Sci. Malh. Hung.

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175

LIST OF SYMBOLS

(&,v)

1

v, (V,A), &+, V * = (V*,A)

2

g(v) = g(A,V), ~(V) = £(A,V), g(V), ~'(V),

dimV,

A•B,

rank a, N a

3

(U,V) = (U,a,V) = (U,/X,V;~), fu' nat U

4

F(~,v), £u(V), Fu(b,V), gU(~,v), ~n,U(b,V), a*

6

A\B, bX' DO{, S-, ~BA-, [i,'f,X] C(A),

7 ii

iB(A )

16

~2, ~ ( N ) ,

~r (g)

19

~Rn

22

Es , } ( U , V )

23

Ge S / I , J(a),

25

I ( a ) , ¢+, g (A,V)

27

(v) Q ~°(I,G,M;P),

28 Ln(A),

Qn,u(A, V)

29

Xv,(V), (®,a), ( a , v ) ~ (Z~',v')

34

(b ,w)

35

(u,b,v) ~ ( u ' , ~ ' , v ' ) , g~,u(b, v)

~(w,a)

38 43

~g, m v

44

C(S) ,£+

45 46

~, A, M, I

47

g, G(S), ¢9(S),

48

HK

60 + V

49 54 k

~(S), %a' IDa' rra' [[(S), T = T(K:S), ~k , p 2

k

55

S

56

~(s)

6O 62

AB, aB, Ab A n , G£,~(B), AB, A n, aB Ai iE I

65 ~r,~(B)

66 67 68

176 70

AB

74 75

k op, £, ~,

76

~(I,A,A;P) g

80 81 91

+G A i iE I

92

~es ~A

¢~

(a)

95

~ ,~ c~AC~ c~

96 97

rl s c~ a6A

102 (a) r x , X A, (X,~) c~A ~

112 •

,

S(~;y), B((~l,...,~n,X[ . . . , X n ) ¢~AIIX , B(vl, . .,Vn;~l, . . .

' Tu(V)

= ~U(A,V)

113

.,~n ), C(vi,.. .,v n;6l,...,~n )'

SJ-, T z, [A], [Wl,W2,..-,Wn],

v't, uZ

114 115

(b,v ,'0

119

codim S

127

a*

X[A], X[x], (xj~) (S,~), [S n In e D}, (f,(X*,%~*))

128 129 130 136

~U (V)' B(xl,...,Xn;Yl,...,yn)' B(xl,...,Xn)(a)

'O.c(a), 7

137

(~, "r)

W(~), "~(A), ~ ,

139 'r

144 145

p(ul, . . .,Un) (a), ~c(a)

<~), ~(M+), M+, r*, ~*

153

N(w)

154

¢y ~ ' ,e--lira

156

~J(Ul.... ~n )

J(~l ..... ~n )' J

160

~(v)

161

~(~)

163

177

INDEX

adjoint of a linear transformation, adjoint of a semilinear affine transformation, algebra,

35

54

46

annihilator,

60

atomic ring,

69

basis,

5

transformation,

3

bilinear

form, 4

biorthogonal

sets, 81

bitranslation,

55

Boolean ring, 46 Cauchy net, 129 center, 45 centralizer, centralizer

48 of a primitive

characteristic,

circle composition, codimension, compatible

ring,

154

16 62

119 (topology-with

complement,

the group structure),

3

complete direct sum, 96 complete

lattice,

125

complete uniform space, completely

O-simple

129

semigroup,

completion of a uniform space, congruence

71 129

(induced by), 54

conjugate of a linear transformation, conjugate of a semilinear conjugate contains

space,

2

small sets,

converges

(a net

dense extension,

133 ), 129

54

dense set of linear transformations, dense uniform subspace, diagonal,

128

dimension,

3

6

transformation,

129

12

127

178

direct product of rings, 96 direct product of semigroups,

97

direct sum, 91 directed

set, 129

directs

(a binary relation ) ,

discrete

division ring, doubly

129

direct sum, 96 i

transitive,

12

dual pair, 4 dual space,

2

endomorphism, essential

3

extension,

60

eventually

in ( n e t ) ,

extension,

54

129

external direct sum, 96 faithful module,

71

finite intersection

property,

finite topology of

X A, 113

finite topology of

~u(V),

finite topology of

V*,

general

linear group,

generalized generates

homothetic hyperplane,

29 62

), 81

set, 3

Green's relations, homogeneous

136

I13

inner automorphism,

(a set

generating

132

75

component,

94

transformation,

54

115

ideal, 6 idealizer,

ii

idempotent

ideal,

identity

66

relation,

in (net

a set),

128 129

independent

family of subrings,

91

independent

system of primitive

ideals,

induced by a semilinear induced congruence,

54

inner automorphism,

48

inner bitranslation,

55

isomorphism,

41

163

179 inner left (right) translation, 55 inverse limit, 156 inverse system of topological modules,

156

invertible matrix, 53 irreducible module, 71 Isomorphic dual pairs, 38 isomorphic vector spaces, 34 Jacobson radical, 163 large ideal, 60 left annihilator,

19,66,91

left ideal, 27 left ~-module,

73

left row-independent matrix, 76 left socle, 69 left socle Lopology, 147,149 left Lranslation, 55 left vector space, I linear form, 2 linear transformation, linear variety,

2

132

linearly compact module,

132

linearly independent set, 3 linearly topologized module,

131

locally matrix ring, 25 A X I-matrix, 76 maximsl dense extension, 54 maximal subgroup, 25 minimal left (right, two-sided) ideal, 67 modular left identity, 141 modular right ideal, 141 multiplication induced by a scalar, 44 n-fold transitive,

12

natural image, 4 natural isomorphism of

V

into

net, 129 nilpotent element, 74 nilpotent ideal, 66 nondegenerate bilinear form, 4 null space, 3

V**, 26

180

open function, 126 order isomorphism,

23

orthogonal set of idempotents,

15

orthogonal sum, 92 pair of dual vector spaces, 4 permutable translations,

61

prime ideal, 71 prime ring, 71 primitive ideal, 162 primitive idempotent, 81 primitive regular semigroup, 97 primitive ring, 71 principal factor, 27 product of relations, 75 product of topological spaces, 112 product topology, 113 projection homomorphism, 96 proper ideal, 32 quasi-inner sutomorphism, 62 range, 3 rank, 3 Rees matrix ring, 76 Rees matrix semigroup, 29 Rees quotient, 27 regular element (semigroup, ring), 19 relative uniformity,

129

relativization of a uniformity, ~-homomorphism,

129

75

rlght annihilator, 19,66,91 right column-independent matrix, 76 rlght ideal 27 rlght R-module,

70

right socle, 69 rlght socle topology, 147,149 right translation, 55 rlght vector space, 2 rlng of endomorphisms,

3

rl-semigroup, 85 ~-isomorpbic modules, ~-isomorpbism,

75

75

181

r-maximal ring, 86 r-maximal semigroup, 86 ~-module,

70

row finite matrix, 52 ~-suhmodule,

70

sandwich matrix, 29 scalsr multiplication,

2

scalars, 2 semidirect product, 50 semlgroup, 3 aemlgroup of endomorphisms,

3

semlgroup socle, 73 semilinear automorphism, 34 semilinear isomorphism of dual pairs, 38 semilinear isomorphism of vector spaces, 34 semilinear transformation,

34

semiprime ring, 66 semiprime ideal, 71 semiprime semigroup, 67 semisimple ring, 163 simple ring, 19 socle, 68 split extension, 51 subdirect sum (product), 96 subspace, 3 subvariety, 132 sum of subrings~ 68 topological group, 114 topological module, 131 topological ring, 137 topological vector space, 115 topologically simple ring, 162 topology of s uniformity,

128

topology of uniform convergence, 144 total subspace, 4 transitive,

12

translation, 54 translational hull, 55 t-subspace, 4

182

uniform isomorphism,

129

uniform semigroup (ring), 63 uniform space~ 128 uniform subspace, 129 uniform topology, 144 uniformity,

128

uniformly continuous function, 129 uniformly equivslent uniform spsces, 129 uniformity of uniform convergence,

144

unique addition, 45 U-topology,

113

vectors, 2 V-topology,

113

weskly reductive semigroup, 54 weskly topologized vector spsce, 131 zero ring, 19

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