3_xxx_jayaram.dvi EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 2, No. 4, 2009, (508-519) ISSN 1307-5543 – www.ejpam.com π-Modules C. Jayaram The University of the West Indies, Department of mathematics, P.O. Box 64, Bridgetown, BARBADOS. Abstract. In this paper we characterize π-modules. Next we establish some equivalent con- ditions for an almost π-module to be a π-module. 2000 Mathematics Subject Classifications: Primary 05C38, 15A15; Secondary 05A15, 15A18 Key Words and Phrases: π-module, almost π-module, multiplication module, π-ring, almost π-ring, multiplication ideal and quasi-principal ideal. 1. Introduction Throughout this paper R denotes a commutative ring with identity and all modules are unital R-modules. L(R) denotes the lattice of all ideals of R. Throughout this paper M denotes a unital R-module. In this paper we introduce and study the concepts of π-module and almost π-module. In Section 3, we prove that a faithful R-module M is a π-module if and only if R is a π-ring and M is a multiplication module if and only if every cyclic submodule of M is of the form I M , where I is a finite product of Email address: jayaram. hillumu� avehill.uwi.edu http://www.ejpam.com 508 c© 2009 EJPAM All rights reserved. C. Jayaram / Eur. J. Pure Appl. Math, 2 (2009), (508-519) 509 quasi-principal prime ideals of rank less than or equal to one (see Theorem 1). Using these results, we establish some equivalent conditions for an almost π-module to be a π-module (see Theorem 2). 2. Basic notions For any x ∈ R, the principal ideal generated by x is denoted by (x). Recall that an ideal I of R is called a multiplication ideal, if for every ideal J ⊆ I , there exists an ideal K with J = KI . Multiplication ideals have been extensively studied - for example, see [2], [5] and [6]. An ideal I of R is called a quasi-principal ideal [16, Exercise 10, Page 147] (or a principal element of L(R) [19]) if it satisfies the following identities (i) (A∩ (B : I))I = AI∩B and (ii) (A+BI) : I = (A : I)+B, for all A, B ∈ L(R). It is well known that an ideal I of R is quasi-principal if and only if it is finitely generated and locally principal if and only if it is a finitely generated multiplication ideal [8, Theorem 3]. R is a π-ring if every principal ideal is a finite product of prime ideals of R. R is an almost π-ring if RP is a π-ring, for every maximal ideal P of R. π-rings have been extensively studied - for example, see [13], [15] and [17]. By a special principal ideal ring, we mean a principal ideal ring R with exactly one prime ideal P 6= R, Pn = (0) for some positive integer n, so the only ideals of R are R, P, P2, ..., Pn = (0). A submodule N of M is proper if N 6= M . For any two submodules N and K of M , the ideal {a ∈ R | aK ⊆ N} will be denoted by (N : K). Thus (O : M) is the annihilator of M . M is said to be a faithful module if (O : M) is the zero ideal of R. M is said to be a multiplication module [9] if every submodule of M is of the form I M , for some ideal I of R. A submodule N of M is said to be a multiplication submodule if for every submodule N1 ⊆ N , there exists an ideal J of R such that N1 = JN . An R-module M is said to be locally cyclic if MP is a cyclic RP-module for all maximal ideals P of R. A proper submodule N of M is said to be a maximal submodule, if it is not properly C. Jayaram / Eur. J. Pure Appl. Math, 2 (2009), (508-519) 510 contained in any other proper submodule of M . A proper submodule N of M is a prime submodule, if for any r ∈ R and m ∈ M , rm ∈ N implies either m ∈ N or r ∈ (N : M). A proper submodule N of M is a primary submodule, if for any r ∈ R and m ∈ M , rm ∈ N implies either m ∈ N or rn ∈ (N : M) for some positive integer n. By a minimal prime submodule over a submodule N of M (or a prime submodule minimal over N), we mean a prime submodule which is minimal in the collection of all prime submodules containing N . Minimal prime submodules over the zero submodule are simply called the minimal prime submodules. It is well known that maximal submodules and prime submodules exist in multiplication modules (for details, see [11]). It is well known that if M is a faithful multiplication R-module and P is a prime ideal of R such that M 6= PM , then PM is a prime submodule of M and every prime submodule of M is of the form PM for some prime ideal P of R (see [11, Corollary 2.11]). Also if M is a faithful and finitely generated multiplication R-module and P is a prime ideal of R, then by [11, Theorem 3.l], PM is a proper prime submodule of M . Further PM is minimal over a submodule N of M if and only if P is minimal over the ideal (N : M) of R. By a multiplicative lattice we mean a complete lattice L on which there is defined a commutative, associative multiplication which distributes over arbitrary joins (i.e., a(∨b α ) = ∨ α ab α ) and has compact greatest element 1 as a multiplicative identity [1]. 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. A principally generated, compactly generated modular multiplicative lattice is called an r-lattice. An r-lattice L is said to be a π-lattice [1] if L is generated by a set S of elements (not necessarily principal) each of which is a finite product of prime elements. It should be mentioned that every principal ideal of R is quasi-principal and hence L(R), the lattice of all ideals of R, is an r-lattice. Note that if R is a π-ring, then L(R) is a π-lattice. For general background and terminology, the reader is referred to [16] and [20]. C. Jayaram / Eur. J. Pure Appl. Math, 2 (2009), (508-519) 511 3. π-modules In this section, we characterize π-modules. Next we establish several equivalent conditions for an almost π-module to be a π-module. We shall begin with the following definition. Definition 1. An R-module M is said to be a π-module if every proper cyclic submodule N of M is of the form I M, where I is a finite product of prime ideals of R. Definition 2. An R-module M is said to be an almost π-module if for any maximal ideal P of R, the RP -module MP is a π-module. Observe that π-rings and cyclic modules over π-rings are examples of π-modules. Almost π-rings are almost π-modules. Again note that π-modules are almost π- modules, but the converse need not be true. Lemma 1. Suppose M is a π-module. Then M is a multiplication module. Proof. The proof of the lemma follows from [11, Proposition 1.1]. Lemma 2. Suppose M is a faithful π-module. Then (i) R contains only finitely many minimal prime ideals of R. (ii) M contains only finitely many minimal prime submodules. (iii) M is finitely generated. Proof. (i). As M is a faithful π-module, the zero ideal is a finite product of prime ideals and hence R contains only finitely many minimal prime ideals. (ii). By Lemma 1, M is a multiplication module. As M is faithful, it follows that every minimal prime submodule is of the form PM for some minimal prime ideal P of R. So by (i), M contains only finitely many minimal prime submodules. (iii). The result follows from (ii) and [11, Theorem 3.7]. C. Jayaram / Eur. J. Pure Appl. Math, 2 (2009), (508-519) 512 Lemma 3. Suppose M is a faithful π-module. Then every proper cyclic submodule of M has only finitely many minimal primes. Proof. Let x ∈ M . As M is a π-module, by definition, Rx = P1P2 · · · PnM for some prime ideals P1, P2, · · · , Pn of R. Let N be a prime submodule minimal over Rx . Note that by Lemma 1 and Lemma 2, M is a faithful and finitely generated multiplication R-module. So N = PM for some prime ideal P of R. As Rx ⊆ N , by [11, Theorem 3.1], it follows that Pi ⊆ P for some i, so Rx ⊆ Pi M ⊆ PM = N . As M is a faithful and finitely generated multiplication R-module, it follows that Pi M is a prime submodule and hence Pi M = N . Therefore Rx has only finitely many minimal primes. This completes the proof of the lemma. Lemma 4. Suppose M is a faithful cyclic R-module. Then R is a π-ring if and only if M is a π-module. Proof. The proof of the lemma follows from [14, Lemma 6] and [11, Theorem 3.1]. Lemma 5. Suppose M is a faithful and finitely generated multiplication R-module. Then M is an almost π-module if and only if R is an almost π-ring. Proof. Let P be a maximal ideal of R. Consider the RP -module MP . As M is a finitely generated faithful multiplication R-module, it follows that MP is a faithful cyclic RP - module. So by Lemma 4, MP is a π-module if and only if RP is a π-ring. Therefore R is an almost π-ring if and only if M is an almost π-module. This completes the proof of the theorem. Lemma 6. Let M be a faithful π-module. If N is a minimal prime submodule, then N is a multiplication submodule. C. Jayaram / Eur. J. Pure Appl. Math, 2 (2009), (508-519) 513 Proof. Note that by Lemma 1 and Lemma 2, M is a faithful and finitely generated multiplication module. Suppose N is a minimal prime submodule. Then N = PM for some minimal prime ideal P of R. Suppose Rx ⊆ PM for some x ∈ M . As M is a π-module, it follows that Rx = I M , where I = P1P2...Pn and Pi ′s are prime ideals of R. Since Rx ⊆ PM , by [11, Theorem 3.1], it follows that Pi ⊆ P for some i. As P is a minimal prime ideal, it follows that P = Pi. Therefore Rx = J(PM) = JN for some J ∈ L(R). Consequently, N is a multiplication submodule. Lemma 7. Let M be a faithful π-module. If N is a prime submodule minimal over a cyclic submodule of M, then N is either minimal or a multiplication submodule with rankN = 1. Proof. Observe that M is a faithful and finitely generated multiplication module. Suppose N is a prime submodule minimal over a cyclic submodule of M . Then N = PM for some prime ideal P of R. Suppose PM is non-minimal. Then P is non-minimal. Let P0 ⊇ P be a maximal ideal of R. Suppose PM is minimal over a cyclic submodule Ry of M . Then by [18, Lemma 1.4], P is minimal over a quasi-principal ideal (Ry : M) of R. Note that by Lemma 5, R is an almost π-ring. Therefore by [12, Theorem 46.8 and Corollary 46.10, Page 576-577], for every maximal ideal Q of R, RQ is either a π-domain or a special principal ideal ring. As RP0 is a π-domain and PP0 is a prime minimal over a non-zero principal element of RP0 , by [15, Theorem 4.2 and Corollary 4.3], PP0 is principal and rank P = 1. Again note that P is locally principal. Observe that by [12, Corollary 46.9, Page 577], every prime ideal contains a unique minimal prime ideal. Let P ′ be a minimal prime ideal contained in P. Then P ′P = P ′ locally and hence globally. Now we show that PM is a multiplication submodule. Suppose Rx ⊆ PM for some x ∈ M . As M is a π-module, it follows that Rx = I M , where I = P1P2...Pn and Pi ′s are prime ideals of R. Since Rx ⊆ PM , it follows that Pi ⊆ P, for some i. So either P ′ = Pi or P = Pi. Therefore Rx = J(PM) = JN for some J ∈ L(R). C. Jayaram / Eur. J. Pure Appl. Math, 2 (2009), (508-519) 514 As M is a π-module, it follows that N = PM is a multiplication submodule. Since rank P = 1, by [11, Theorem 3.1], rank N = 1. This completes the proof of the lemma. Lemma 8. Let M be a faithful π-module. Then every cyclic submodule is a finite inter- section of primary submodules. Proof. Let Rx be a cyclic submodule of M and let I = (Rx : M). As M is a faithful π-module, by Lemma 3, it follows that Rx has only finitely many minimal primes. As M is a faithful and finitely generated multiplication module, it follows that every prime submodule is of the form PM for some prime ideal P of R. Let Pi ′s for i = 1, 2, ..., m be the distinct prime ideals of R such that P1M , P2M , ..., PmM are the distinct prime submodules which are minimal over Rx . It can be easily seen that a prime ideal P of R is minimal over I if and only if P = Pi for some i. As R is an almost π-ring, by [12, Corollary 46.10, Page 577], it follows that the non-maximal minimal primes are unbranched and idempotent. By Lemma 6 and Lemma 7, each Pi M is a multiplication submodule and rank Pi ≤ 1 for i = 1, 2, ..., m. Again by [11, Theorem 3.1] and [18, Lemma 1.4], each Pi is a multiplication ideal. Without loss of generality, assume that P1, P2, ..., Ps are the rank one multiplication prime ideals, Ps+1, Ps+2, ..., Ps+t are the non maximal minimal primes and Ps+t+1, Ps+t+2, ..., Pm are the minimal primes which are also maximal. Since P1, P2, ..., Ps are the rank one multiplication prime ideals, by [5, Theorem 3], these are quasi-principal ideals. Therefore by [3, Theorem 2.2], there exist positive integers ni ′s for i = 1, 2, ..., s, such that I ⊆ P ni i and I 6⊆ P ni+1 i . Since each RPi (s+ t + 1 ≤ i ≤ m) is a special principal ideal ring, there exist positive integers n j ′s for (s + t + 1 ≤ j ≤ m) such that IPj = (P n j j )Pj . Observe that by [5, Corollary] and [6, Lemma 1], the powers of Pi (1 ≤ i ≤ m) are multiplication Pi- primary ideals. Let J = P n1 1 ∩ P n2 2 ∩ ...∩ Pns s ∩ Ps+1 ∩ ...∩ Ps+t ∩ P ns+t+1 s+t+1 ∩ ...∩ Pnm m . Now we claim that I = J . Let Q be a maximal prime ideal of R. If Pj ⊆ Q for some j ∈ {s + 1, s + 2, ...., s+ t}, then IQ = JQ = 0Q as RQ is a π-domain. Without loss of C. Jayaram / Eur. J. Pure Appl. Math, 2 (2009), (508-519) 515 generality, assume that P1, P2, ..., Pt ⊆Q for (1 ≤ t < s) and Pj 6⊆Q for (t+1 ≤ j ≤ s). Note that RQ is a π-domain and IQ is a non zero principal ideal of RQ. Therefore by [15, Theorem 4.2 and Corollary 4.3], IQ is a finite product of the rank one principal prime ideals minimal over it. Again using Theorem 3 of [7], it can be easily shown that IQ = (P1 n1)Q ∩ (P2 n2)Q ∩ ... ∩ (Pt nt )Q. Therefore IQ = JQ since (Pj n j) Q = RQ for (t+1≤ j ≤ s) and (Pk)Q = RQ for (s+1 ≤ k ≤ m). If Pj ⊆Q for (s+t+1 ≤ j ≤ m), then IQ = JQ. This shows that IQ = JQ for all maximal prime ideals Q containing I . Further, if I 6⊆ Q, then IQ = JQ = RQ. Consequently, I = J and hence Rx = I M = J M . Since J is a finite intersection of primary ideals, by [11, Theorem 1.6] and [21, Corollary 1], Rx is a finite intersection of primary submodules. This completes the proof of the lemma. Theorem 1. Suppose M is a faithful R-module. Then the following statements on M are equivalent: (i) M is a π-module. (ii) R is a π-ring and M is a multiplication module. (iii) Every cyclic submodule of M is of the form I M, where I is a finite product of quasi-principal prime ideals of rank less than or equal to one. Proof. (i)⇒(ii). Suppose (i) holds. Then M is a faithful and finitely generated multiplication module. By Lemma 5, R is an almost π-ring. By [13, Theorem 6], it is enough if we show that every prime ideal of R of rank less than or equal to one is finitely generated. Note that if P is a minimal prime ideal of R, then PM is a minimal prime submodule, so by [14, Lemma 7], Lemma 6 and Lemma 8, PM is a finitely generated multiplication submodule. Therefore by [18, Lemma 1.4], P = (PM : M) is a finitely generated multiplication ideal. Let P be a rank one prime ideal. As R is an almost π-ring, it follows that P is locally principal. So by [14, Lemma 6], PM is locally cyclic and hence by [14, Lemma 7] and Lemma 8, PM is a finitely generated C. Jayaram / Eur. J. Pure Appl. Math, 2 (2009), (508-519) 516 multiplication submodule. Consequently, P is a finitely generated multiplication ideal. Therefore R is a π-ring. (ii)⇒(iii). Suppose (ii) holds. Let x ∈ M . As M is a multiplication module, it follows that Rx = (Rx : M)M . Since R is a π-ring, it follows that R contains only finitely many minimal prime ideals and so M contains only finitely many minimal prime submodules. So by [11, Theorem 3.7], M is finitely generated. Again by [18, Lemma 1.4], (Rx : M) is a quasi-principal ideal of R (i.e., a principal element of R). As R is a π-ring, it follows that L(R) is a π-lattice, so by [4, Theorem 2], (Rx : M) is a finite product of quasi-principal prime ideals of rank less than or equal to one. Therefore (iii) holds and (iii)⇒(i) follows from the definition. This completes the proof of the theorem. Theorem 2. Suppose M is a faithful R-module. Then the following statements on M are equivalent: (i) M is a π-module. (ii) M is an almost π-module in which every finitely generated multiplication sub- module is a finite intersection of primary submodules. (iii) M is an almost π-module in which every cyclic submodule is a finite intersection of primary submodules. (iv) M is finitely generated and an almost π-module in which every prime submodule of rank less than or equal to one is finitely generated. (v) M is a multiplication module in which every minimal prime submodule is a finitely generated multiplication submodule and every non minimal prime submodule contains a non minimal finitely generated multiplication prime submodule. Proof. (i)⇒(ii). Suppose (i) holds. Clearly, M is an almost π-module. Let N be a finitely generated multiplication submodule. As M is a faithful and finitely generated multiplication module, by [18, Lemma 1.4], (N : M) is a principal element of L(R). C. Jayaram / Eur. J. Pure Appl. Math, 2 (2009), (508-519) 517 So by Theorem 1 and [13, Theorem 6], (N : M) is a finite intersection of primary ideals and hence by [11, Theorem 1.6] and [21, Corollary 1], N = (N : M)M is a finite intersection of primary submodules. (ii)⇒(iii) is obvious. (iii)⇒(iv). Suppose (iii) holds. By (iii) and Lemma 1, M is locally cyclic. So by [14, Lemma 7], M is a finitely generated multiplication module. Let N be a prime submodule of rank less than or equal to one. As M is a faithful and finitely generated multiplication module, N = PM for some prime ideal P of rank less than or equal to one. By the proof of (i)⇒(ii) of Theorem 1, P is finitely generated and hence N is finitely generated. Therefore (iv) holds. (iv)⇒(v). Suppose (iv) holds. By (iv), M is finitely generated and locally cyclic. So by [9, Proposition 5], M is a finitely generated multiplication module. So by Lemma 5, R is an almost π-ring. As M is a faithful and finitely generated multiplication module, by hypothesis and [11, Theorem 3.1], every prime ideal of rank less than or equal to one is finitely generated. So by [13, Theorem 6], the minimal prime ideals of R are quasi-principal ideals and every non minimal prime ideal contains a non minimal quasi-principal prime ideal. Now the result follows from [18, Lemma 1.4] and [11, Theorem 3.1]. Therefore (v) holds. (v)⇒(i). Suppose (v) holds. By hypothesis, the minimal prime submodules are finitely generated, so M contains only finitely many minimal prime submodules and hence M is finitely generated (for details, see [10, Theorem 2]). As M is a faithful and finitely generated multiplication module, by (v), [11, Theorem 3.1] and [18, Lemma 1.4], the minimal prime ideals of R are quasi-principal and every non minimal prime ideal contains a non minimal quasi-principal prime ideal. Now the result follows from Theorem 1 and [13, Theorem 6]. 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