Algebra univers. 45 (2001) 07 – 13 0002–5240/01/010007 – 07 $ 1.50 + 0.20/0 © Birkh¨auser Verlag, Basel, 2001
2-Join decomposition lattices C. Jayaram
Abstract. In this paper we characterize principal element lattices and Dedekind domains in terms of 2-join decomposition lattices.
A commutative ring R with identity is called a general ZPI-ring [5, page 469] if every ideal is a finite product of prime ideals. For various characterizations of general ZPIrings the reader is referred to [13], [14] and [15]. Butts and Gilmer [3, Theorem 11 and Corollary 6] have shown that R is a general ZPI-ring if and only if every ideal is a finite intersection of powers of prime ideals. This result has been generalized by Johnson [12]. Johnson [12, Theorem 7] has proved that R is a general ZPI-ring if and only if every doubly generated ideal is a finite intersection of powers of prime ideals. Butt’s and Gilmer’s result has been extended to multiplicative lattices in [9]. In fact in [9], it is shown that a principally generated C-lattice is a principal element lattice if and only if every element is a finite meet of prime power elements. In this paper we introduce 2-join decomposition lattices and characterize principal element lattices in terms of 2-join decomposition lattices which extend a result of Johnson [12, Theorem 7] and Theorem 4 of [9] (see Theorem 1 and Theorem 2). Finally, it is established that a principally generated C-lattice L (which is not a two element chain) is a finite direct product of proper Dedekind domains if and only if L is a reduced 2-join decomposition lattice in which the maximal elements are non-idempotent. An element e of a multiplicative lattice L is said to be principal if it satisfies the dual identities (i) a ∧ be = ((a : e) ∧ b)e and (ii) (a ∨ be) : e = (a : e) ∨ b. By a C-lattice we mean a (not necessarily modular) complete multiplicative lattice, with least element 0 and compact greatest element 1 (a multiplicative identity), which is generated under joins by a multiplicatively closed subset C of compact elements. In a principally generated C-lattice, principal elements are compact [1, Theorem 1.3]. Throughout this paper we assume that L is a C-lattice and R is a commutative ring with identity. C-lattices can be localized. For any prime element p of L, Lp denotes the Presented by Professor Boris M. Schein. Received March 2, 1999; accepted in final form July 10, 2000. 1991 Mathematics Subject Classification: 06F10, 06F05, 06F99, 13A15. Key words and phrases: Prime power element, general ZPI-ring, 2-join decomposition lattice.
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localization at F = {x C | x 6 ≤ p}. For basic properties of localization, the reader is referred to [8]. Principal elements were introduced into multiplicative lattices by R. P. Dilworth [4]. A multiplicative lattice L in which every element is principal is called a principal element lattice. Similarly, L is said to be an almost principal element lattice if Lm is a principal element lattice for every maximal element m of L. For various characterizations of almost principal element lattices and principal element lattices, the reader is referred to [2], [6] and [7]. It is well known that if L is principally generated, then L is a principal element lattice if and only if every prime element principal. For general background and terminology, the reader may consult [1] and [7]. 1. Prime power elements in C-lattices In this section we study prime power elements in C-lattices. We shall begin with the following definition. DEFINITION 1.1. A principally generated C-lattice L is said to be a 2-join decomposition lattice if there exists a multiplicatively closed set S of (not necessarily principal) elements such that S generates L under joins and for every x, y S, x ∨ y is a finite meet of prime power elements. Principal element lattices and Noether lattices [4] in which every primary element is a power of its radical are examples of 2-join decomposition lattices. If R satisfies property (D) [12], then L(R), the lattice of all ideals of R is a 2-join decomposition lattice. Throughout this section, we assume that L is a 2-join decomposition lattice. Let S denote a multiplicatively closed set of (not necessarily principal) elements of L which generate L under joins such that for every x, y S, x ∨ y is a finite meet of prime power elements. We now prove some useful lemmas. LEMMA 1.2. Let x L be a principal element. Then x has only finitely many minimal primes over x. Proof. Observe that every element a S has only finitely many minimal primes over a. Since x is compact, it follows that x = ∨ni=1 xi (xi0s S). It is enough if we show that every minimal prime p over x is a minimal prime over xi for some i. Let p be a minimal prime over x. By Theorem 1.2 of [1], xp is join irreducible in Lp , so xp = (xi )p for some i. As p is a minimal prime over x, it follows that (xi )p = xp is p-primary [8, Property 0.5]. Hence p is a minimal prime over xi . This completes the proof of the lemma. ¨ LEMMA 1.3. Let m be a prime element of L. Then Lm is a 2-join decomposition lattice.
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Proof. Let H be the set of all principal elements of L and let Hm = {am | a H }. As L is principally generated, it follows that Hm generates Lm under joins. It is enough if we show that for any am , bm Hm , am ∨ bm is a finite meet of prime power elements in Lm . Let am , bm Hm . Then am = xm and bm = ym for some x, y S. Note that am ∨ bm = xm ∨ ym = (x ∨ y)m and x ∨ y = ∧ni=1 piαi for some prime elements pi0s L. So am ∨ bm = ∧{(pi )αmi |pi ≤ m}. This completes the proof of the lemma. ¨ LEMMA 1.4. Let m be a prime element of L. Then either m = m2 or mm is principal in Lm . Proof. Suppose m 6 = m2 . Choose any principal element a ≤ m such that a 6 ≤ m2 . Then there exists x1 S such that x1 ≤ a and x1 6 ≤ m2 . Let y ≤ m be any principal element of L. Then ym = (x2 )m for some x2 S. Note that x1 ∨ (x2 )2 = ∧ni=1 piαi for some prime elements pi0s of L. Since x1 ∨ (x2 )2 ≤ m, it follows that pi ≤ m for some i, say p1 ≤ m. As x1 6 ≤ m2 , we have αi = 1 for all pi ≤ m. Therefore (x1 )m ∨ (y 2 )m = (x1 ∨ (x2 )2 )m = ∧{pi | pi ≤ m}. Since each pi is a prime element, we have (x1 )m ∨ ym = (x1 )m ∨ (x2 )m = ∧{pi | pi ≤ m}. Hence by Theorem 1.4 of [1], ym ≤ (x1 )m ≤ am and therefore mm = am which is principal in Lm . ¨ LEMMA 1.5. Let p be a minimal prime over a principal element y L. Then pn is p-primary for all n Z + . Further if q < p is a prime element, then q < pn for all n Z + . Proof. Let n Z + and let r, s L be principal elements such that rs ≤ pn and s 6 ≤ p. Since r and y are principal elements, it follows that r = ∨ni=1 xi (xi0s S) and ypn = zp for αi some z S. We claim that each xi ≤ p n . Let i {1, 2, . . . , n}. Note that xi ∨ z = ∧m i=1 pi for some prime elements pi0s L. Since p is a minimal prime over xi ∨ y n , it follows α that (xi ∨ y n )p = (xi ∨ z)p = ppj where p = pj for some j {1, 2, . . . , m}. But αj n n n pp = (xi ∨y )p = (xi s∨y )p ≤ pp . So αj ≥ n and pαj ≤ pn . Therefore xi ≤ pαj ≤ pn and hence r ≤ pn . This shows that pn is p-primary for all n Z + . Suppose q < p is a prime element. If p = p 2 , then we are through. Suppose p 6= p2 . Then by Lemma 1.4, pp is principal in Lp . Again since pp is a non minimal principal prime element of Lp , by ¨ Lemma 1.4 of [2], q = qp < ppn = pn for all n Z + . LEMMA 1.6. Let p be a prime element of L. Then pn is p-primary for all n Z + and if q < p is a prime element of L, then q < p n for all n Z + . Proof. If p = p 2 , then we are through. Suppose p 6= p2 . Then by Lemma 1.4, p = ap for some principal element a L. So p is a minimal prime over a. Now the result follows from Lemma 1.5. ¨ LEMMA 1.7. For any x, y S, x ∨ y is a finite meet of primary elements of L.
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Proof. The proof of the lemma follows from Lemma 1.6.
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LEMMA 1.8. Let p be a non-idempotent prime element of L. Then Lp is a principal element lattice. Proof. By Lemma 1.4, pp is principal in Lp . Suppose pp = ap for some principal element a L. If p is minimal, then we are through. Suppose p is a non-idempotent nonminimal prime element of L. Since p is principal in Lp , by Lemma 1.6, every p-primary element is p n for some n Z + . We show that if q < p is a prime element, then q = 0p . Let q < p be a prime element. Let b ≤ q be any principal element of L. Note that (ab)p = xp for some x S and by Lemma 1.7, x is a finite meet of primary elements of L. Let x = q1 ∧ · · · ∧ qn be a normal primary decomposition of L. Suppose qi ≤ p for i = 1, 2, . . . , k and qj 6 ≤ p for j = k + 1, . . . , n. Then xp = ∧ki=1 {(qi )p }. By Lemma 1.6, √ √ we can assume that qi < p for i = 1, 2, . . . , k. As qi < p for i = 1, 2, . . . , k, it √ follows that a 6 ≤ qi , so b ≤ ∧ki=1 qi and hence bp = xp in Lp . So bp ap = xp = bp in Lp . Again by Theorem 1.4 of [1], bp = 0p and hence qp = 0p . This shows that Lp is a one dimensional principal element lattice. Thus Lp is a principal element lattice and the proof is complete. ¨ LEMMA 1.9. Let p = p 2 be a prime element of L and let x be a compact element of L. Suppose y ≤ x ≤ p for some y S. If p is a minimal prime over y, then p is a minimal prime element of L. Proof. Suppose p is a minimal prime over y. Since y S, it follows that y is a finite meet of prime power elements. As p is a minimal prime over y, it follows that yp = ppα for some α Z + . But p = p2 , so yp = pp and hence xp = pp . Since pp is a compact and idempotent element of Lp , we have pp = 0p [1, Theorem 1.4]. Therefore p is a minimal prime. ¨ Following [11], we say that a multiplicative lattice L0 satisfies the union condition on primes if for any set p1 , p2 , . . . , pn of primes in L0 and any a L0 with a 6≤ p1 , . . . , pn , there exists a principal element e ≤ a with e 6≤ p1 , p2 , . . . , pn . LEMMA 1.10. Suppose L satisfies the union condition on primes. If p = p2 is a prime element of L, then Lp is a two element chain. Proof. Suppose p is an idempotent prime element of L. Note that each x S has only finitely many minimal primes. Suppose p is not a minimal prime. Let p1 , p2 , . . . , pn be the minimal primes. By hypothesis, there exists a principal element x ≤ p such that x 6 ≤ pi for i = 1, 2, . . . , n. As x is compact, it follows that x = ∨ki=1 xi for some x1 , x2 , . . . , xk S. Let H = {qα L | qα ≤ p is a minimal prime over xi for some i {1, 2, . . . , k}}. Note that
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H is a finite set and by Lemma 1.9, p 6 ≤ qα for all qα H . Again by the union condition on primes, there exists a principal element y ≤ p such that y 6≤ qα for all qα H . Observe that x ∨ y is not contained in any prime p1 ≤ p of rank ≤ 1. Let q ≤ p be a minimal prime over x ∨ y. If q = q 2 , then (x ∨ y)q = qq (see the proof of Lemma 1.3), so q is a minimal prime, a contradiction. So assume that q 6= q 2 . By Lemma 1.8, rank q ≤ 1, which is again a contradiction. Therefore p is a minimal prime. Again since 0 is a finite meet of prime power elements and p is an idempotent minimal prime, it follows that 0p = pp in Lp . This shows that Lp is a two element chain. ¨ LEMMA 1.11. Suppose every compact element x L is a finite meet of prime power elements. If p = p2 is a prime, then Lp is a two element chain. Proof. The proof of the lemma is similar to that of Lemma 1.10.
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2. General ZPI-rings In this section we characterize principal element lattices and general ZPI-rings in terms of 2-join decomposition lattices. Throughout this section we assume that L is a principally generated C-lattice. THEOREM 2.1. The following statements on L are equivalent: (i) (ii) (iii) (iv)
L is a principal element lattice. L is a 2-join decomposition lattice and every compact element is principal. L is a 2-join decomposition lattice and satisfies the union condition on primes. Every compact element of L is a finite meet of prime power elements.
Proof. (i) ⇒ (ii). Suppose (i) holds. Then every element is a finite meet of prime power elements [9, Theorem 4] and obviously every element is principal. (ii) ⇒ (iii) follows from the fact that a finite join of compact elements is compact. (iii) ⇒ (i). Suppose (iii) holds. Let m be a maximal element of L. If m 6 = m2 , then by Lemma 1.8, Lm is a principal element lattice. If m = m2 , then by Lemma 1.10, Lm is a two element chain. Therefore L is an almost principal element lattice. Again by Lemma 1.2, every principal element has only finitely many minimal primes and hence by [7, Theorem 8], L is a principal element lattice. Therefore (i) holds. (i) ⇒ (iv) is well known [9, Theorem 4]. (iv) ⇒ (i). Suppose (iv) holds. Then L is a 2-join decomposition lattice. So by Lemma 1.8 and by Lemma 1.11, L is an almost principal element lattice. Again by (iv), every principal element is a finite meet of prime power elements and hence every principal element has only finitely many minimal primes. Therefore L is a principal element lattice. This completes the proof of the theorem. ¨
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Using Theorem 2.1, we now characterize general ZPI-rings in terms of 2-join decomposition lattices. Let L(R)∗ be the lattice of all proper ideals of R. The following Theorem 2.2 is a generalization of Theorem 7 of [12]. THEOREM 2.2. The following statements on R are equivalent: (i) (ii) (iii) (iv)
R is a general ZPI-ring. Every doubly generated ideal in L(R)∗ is a finite intersection of prime powers. L(R)∗ is a 2-join decomposition lattice. L(R)∗ is a principal element lattice.
Proof. (i) ⇔ (ii) follows from Theorem 7 of [12], (ii) ⇒ (iii) by definition and (iii) ⇔ (iv) follows from Theorem 2.1 and the fact that L(R)∗ satisfies the union condition on prime ideals. (iv) ⇒ (i) follows from Theorem 5 of [10]. This completes the proof of the theorem. ¨ L is said to be reduced if 0 is the only nilpotent element of L. A prime element p L is said to be unbranched if p is the only p-primary element. An element a L is called invertible if a is principal and (0 : a) = 0. A domain L is said to be a Dedekind domain if every element is a finite product of prime elements. It is well known that L is a Dedekind domain if and only if L is a principal element domain [2, Theorem 2.7]. A multiplicative lattice domain is said to be a proper domain if it is not a two element chain. Finally, we prove that L is a finite direct product of proper Dedekind domains if and only if L is a reduced 2-join decomposition lattice in which the maximal elements are non-idempotent. THEOREM 2.3. Suppose L is not a two element chain. Then L is a finite direct product of proper Dedekind domains if and only if L satisfies the following conditions: (i) L is reduced. (ii) Every maximal element of L is non-idempotent. (iii) L is a 2-join decomposition lattice. Proof. Suppose L = L1 × · · · × Ln where each Li is a proper Dedekind domain. Obviously, L is reduced. Since each Li is a Dedekind domain, it follows that each Li is a principal element domain and so L is a principal element lattice. Again by Theorem 3.2 of [2], every maximal element is invertible and hence the maximal elements are nonidempotent. Thus L satisfies the conditions (i), (ii) and (iii). Conversely, assume that L satisfies the conditions (i), (ii) and (iii). By (i), the minimal prime elements are unbranched. So by (ii) the maximal elements are non-minimal prime
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elements. By (iii) and Lemma 1.8, L is an almost principal element lattice and hence by [6, Theorem 2], every compact element is principal. Therefore by Theorem 2.1, Lis a principal element lattice. As L is a principal element lattice, the maximal elements are non-minimal principal primes. Again since L is reduced, by [1, Theorem 2.3], it follows that the maximal elements are invertible. Therefore by Theorem 3.2 of [2], L is a finite direct product of proper Dedekind domains. This completes the proof of the theorem. ¨
Acknowledgement The author would like to thank the referee for his helpful comments and suggestions. REFERENCES [1] Anderson, D. D., Abstract commutative ideal theory without chain condition, Algebra univers. 6 (1976), 131–145. [2] Anderson, D. D. and Jayaram, C., Principal element lattices, Czechoslovak Math. Jour. 46 (1996), 99–109. [3] Butts, H. S. and Gilmer, R. W., Jr., Primary ideals and prime power ideals, Canad. Jour. Math. 18 (1966), 1183–1195. [4] Dilworth, R. P., Abstract commutative ideal theory, Pacific Jour. Math. 12 (1962), 481–498. [5] Gilmer, R. W., Multiplicative ideal theory (Marcel Decker, 1972). [6] Jayaram, C. and Johnson, E. W., Almost principal element lattices, Inter. Jour. Math and Math. Sci. 18 (1995), 535–538. [7] Jayaram, C. and Johnson, E. W., Some results on almost principal element lattices, Periodica Mathematica Hungarica. 31 (1995), 33–42. [8] Jayaram, C. and Johnson, E. W., s-prime elements in multiplicative lattices, Periodica Mathematica Hungarica. 31 (1995), 201–208. [9] Jayaram, C. and Johnson, E. W., Primary elements and prime power elements in multiplicative lattices, Tamkang Jour. Math. 27 (1996), 111–116. [10] Johnson, E. W. and Lediaev, J. P., Representable distributive Noether lattices, Pacific Jour. Math. 28 (1969), 561–564. [11] Johnson, E. W. and Lediaev, J. P., Structure of Noether lattices with join principal maximal elements, Pacific Jour. Math. 37 (1971), 101–108. [12] Johnson, E. W., Almost Dedekind rings, Glasgow Math. Jour. 36 (1994), 131–134. [13] Levitz, K. B., A characterization of general ZPI-rings, Proc. Amer. Math. Soc. 32 (1972), 376–380. [14] Levitz, K. B., A characterization of general ZPI-rings II, Pacific Jour. Math. 42 (1972), 147–151. [15] Wood, C. A., On general ZPI-rings, Pacific Jour. Math. 30 (1969), 837–846. C. Jayaram Department of Mathematics University of Botswana Private Bag 00704 Gaborone Botswana e-mail:
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