Matrices and semigroups of ring endomorphisms

Semigroup Forum, Vol. 5 (1972), 160-166.

(0,i) - MATRICES AND SEMIGROUP$ OF RING ENDOMORPHISMS Carlton J. Maxson Communicated by P. A. Grillet

In this paper we investigate semigroups of ring endomorphisms of several classes of rings. As one result we find that the Green relations in the endomorphism semigroup, End R, for a ring R in a given class, are restrictions of those in the transformation semigroup ' f i R "

i.

Introduction. E. S. Ljapin,

[5], in his discussion of the future

of research in semigroups, makes a considerable point of the importance of studying semigroups of endomorphisms of algebraic structures in order to determine properties of the structures

themselves.

appearance of this statement,

Since the

semigroups of endomor-

phisms of vector spaces have been systematically studied by several mathematicians

(see e.g.

[2]) and

these results show that the theory of semigroups can be successfully applied to the study of endomorphisms vector spaces.

of

Following the suggestion of Ljapin, we 160

9 1972 by Springer-Verlag New York Inc.

MAXSON

investigate the semigroups of ring endomorphisms of certain classes of rings.

2.

Products of Cyclic Rings. We first investigate the semigroup of ring endo-

morphisms of a ring R with identity which is a product n of a finite number of cyclic rings, R = H R i. Recall i=l that a ring R i with identity is cyclic if its additive group is cyclic.

Moreover, R i ~ Z, the ring of integers

or R i is isomorphic to some residue class ring, Zn(i), for some positive integer n(i).

If

~i (i) ~m (i) n(i) = Pl "''Pm where the pj are distinct primes, then Zn(i) is ring isomorphic to ZPll(1). x...x Zpmm~ (i)" Thus R is isomorphic to Z x...x Z

x

Z

(*)

x...

x...x Z PI ll

Pllrlj

ro ...x

Z

x...x Z

Ph ~hl.

,

phc~hrh

where, without loss of generality, we order the factors in such a manner that ~iJ ~ ~ik when j ~ k. Let End R denote the semlgroup of ring endomorphlsms of R and EndlR, the subsemigroup of identity preserving ring endomorphisms.

161

For i # k,

MAXSON

End(Z

, Z ~ ) = {0}. PiiJ Pk ks

L E M M A 2.1.

For i = k we have

Let @ e End(Z

, Z P

).

If ~ < B, ~ = 0

P

> 8 then @ = 0 or e @ = e~ where e multiplicative

identities

of Z

and e~ are the D

and Z P

Also,

respectively. P

if ~ e End(Z e' Z), then ~ -= 0 while P

e End(Z,

if

Z ~) then ~ i s th___eezero map or the m o r p h i s m P

preservin~

identities.

Notation:

Since the above identity preserving

morphisms

If

Z 8 + Z e and Z § Z e are of the form x § x P P P

(rood p~) we use "i" to denote them. THEOREM

2.2.

i s semiKroup

Let R be defined isomorphic

as in (*).

to the collection

Then End R of all r x r

h matrices

(where r =

Z

r i) of the form

i=0 m

#ii

-

(**)

[~iJ]

=

~ir 0

"'"

~ir

~roro

" ..

~ro r

ryo

0

...o

q o...o 0

0 162

0

MAXSON

Each A S i = i, 2 h is square matrix i' ,..., a consisting o f 0's and l's; consequently

[$ij ] can be

considered as a (011)-matrix with at most a single 1 i__n_n each column.

Further EndlR is isomorphic t_~o SI, the

subsemigroup o f S consisting o f iOll)-matrices with exactly one 1 in each column.

3,

Products of rigid rings. A ring K is said to rigid if K 2 # (0) and

End K = {0,i} where 1 denotes the identity map on K. Let R = rigid ring. THEOREM 3.1.

n H R. such that, for every i, R. = K, K a i=l I 1 Similar to Theorem 2.2 we obtain If R = --

n H R., R. = K, K ! rigid ring 1 i i=l

then End R ~ S where S is the semigroup o f n x n column monomial

(011)-matrices and EndlR is isomorphic to the

semi~roup of n x n strictly column monomial (0,1)-matrices. We now turn to a determination of the Green relations on End R where R is a ring as described in the above theorem.

We recall that in ~ X '

the full

transformation semigroup on a set X, the Green relations have been characterized in terms of images and equivalence kernels

([i]).

For any ring T, End T is a subsemigroup of J T " We consider the question:

Are the Green relations in

163

MAXSON

End T restrictions of those in J T " question has a negative answer. the field of rational functions over Z 2.; i.e. T = Z2(x).

In general this

For example,

in one indeterminant

The map s : T § T

determined by ks = k, k E T and xa = x but not surjective. sEnd T # i End T.

2

is injective

This implies Ker s = Ker i but Also,

there exist rings T such that

for s, 8 c End T, Im a = Im 8 but However,

let T be

(End T)a ~ (End T)8.

for products of rigid rings we do obtain

an affirmative answer to our question. T H E O R E M 3.2. rigid ring.

Let R =

n H R. w h e r e each R i = K, K a 1 i=l

For ~, B E End R, let ~ correspond to the

m a t r i x A and B to the m a t r i x B.

The following are

equivalent : (a)

Ker s = Ker B,

(b)

A and B have the same non-zero rows,

(c)

s End R = B End R.

T H E O R E M 3.3. theorem,

as the previous

the following are equivalent:

(a)

Im s = I m

(b)

There exists a p e r m u t a t i o n m a t r i x P such that PA=

(c)

4.

Under the same hypothesis

8.

B.

(End R) s = (End R) 8.

A R e p r e s e n t a t i o n Theorem. Several authors have obtained c h a r a c t e r i z a t i o n s

semigroups as e n d o m o r p h i s m semigroups algebraic structures.

G. Gr~tzer 164

of various

[3] shows that a

of

MAXSON

semigroup

S is isomorphic

of some abstract identity.

to an endomorphism

semigroup

algebra A if, and only if, S has an

Closely related to this is the recent work

of Hedrlin and Lambek any semigroup many monoids

[4] in which they establish:

S with identity,

there exist arbitrarily

(i.e., semigroups with identity)

that S is isomorphic

for

M such

to the semigroup of endomorphisms

of M. In this section we give a partial solution corresponding

problem of rings.

to the

That is, using a

procedure

similar to one used in [I], we show that every

semigroup

S can be represented

semigroup,

as a subsemigroup

End R, of ring endomorphisms

of the

of some ring R.

In fact, R can be chosen in a "nice" way. THEOREM 4.1.

Every semigroup

semigroup of ~

S is isomorphic

to

preserving ring endomorphisms

some Boolean ring. (Sketch of proof.)

Let s E S I and define

M(s) ~ (aij(s)), where

(i, j) e S I x S I by 1, if sj = i,

aij (s) = 0, otherwise. : S § M = {M(s)

I s e S I} is a representation

and M can be considered where R =

as a subsemigroup

~ Re, R~ ~ Z2, seA

[A 1 = IS11.

165

of S

of EndlR

of

MAXSON

REFERENCES

[13

Clifford, A. and Preston, G., Algebraic Theory of Semigroups, Amer. Math. Soc. Surveys, Vols I and II, Providence, 1961 and 1967.

[2]

Gluskin, L., Semigroups and rings o_~_fendomorphisms of linear spaces, I and II. Amer. Math. Soc. Translations, Series 2, 45(1965), 105-145.

[3]

Gr~tzer, G., On the endomorphism semigroup o_~f simple algebras. Math. Annalen 170(1967), 334-338.

[4]

Hedrlin, Z. and Lambek, J., How comprehensive is the category o_~fsemigroups? Journal of Algebra 11(1969), 195-212.

[53

Ljapin, E., Semigroups. Amer. Math. Soc. Translations, Vol 3, Providence, 1963.

Mathematics Department Texas A&M University College Station, Texas

77843

Received April 7, 1972 and in revised form, July 24, 1972.

166