9_xxx_jayram.dvi EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 3, No. 5, 2010, 899-902 ISSN 1307-5543 – www.ejpam.com A Note on Prüfer Modules C. Jayaram1,∗and V.C.C. Raju2 1 The University of the West Indies, Department of mathematics, P.O. Box 64, Bridgetown, BARBA- DOS 2 Department of Mathematics, University of Botswana, Gaborone, BOTSWANA Abstract. In this paper we characterize Prüfer modules and Dedekind modules. 2000 Mathematics Subject Classifications: Primary 13C13; Secondary 13C05, 13A15 Key Words and Phrases: multiplication module, Prüfer module, Dedekind module, quasi-principal ideal, quasi-cyclic module. 1. Introduction Throughout this paper R denotes a commutative ring with identity and M denotes a unital R-module. L(R) (L(M)) denotes the lattice of all ideals of R (submodules of 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 [4] if every submodule of M is of the form IM , for some ideal I of R. According to [7], a submodule N of M is called meet-quasi-cyclic (or meet principal in the sense of [1, 3]) if (B ∩ (K : N))N = BN ∩ K for all ideals B of R and for all submodules K of M ; N is called weak-join-quasi-cyclic if (BN) : N = (0 : N) + B for all ideals B of R; N is called join-quasi-cyclic (or join principal in the sense of [1] and [3]) if (K + BN) : N = (K : N) + B for all ideals B of R and for all submodules K of M . N is called quasi-cyclic [7] (or principal in the sense of [1] and [3])if N is both meet-quasi-cyclic and join-quasi-cyclic. Note that quasi-cyclic submodules have been studied in [1], [3] and [7]. For any a ∈ R, the principal ideal generated by a is denoted by (a). 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 . An ideal I of R is called weak join principal if (AI : I) = A+ (0 : I) for all A ∈ L(R). I is called join principal if (A+ BI) : I = (A : I) + B, for all A, B ∈ L(R). An ideal I of R is called a quasi-principal ideal [8, Exercise 10, Page 147] (or a principal element of L(R) [9]) if it satisfies the identities ∗Corresponding author. Email addresses: jayaram. hillumu� avehill.uwi.edu (C. Jayaram), varanasi�mopipi.ub.bw (V. Raju) http://www.ejpam.com 899 c© 2010 EJPAM All rights reserved. C. Jayaram and V. Raju / Eur. J. Pure Appl. Math, 3 (2010), 899-902 900 (i) (A∩ (B : I))I = AI ∩ B and (ii) (A+ BI) : I = (A : I) + B, for all A, B ∈ L(R). Obviously, every quasi-principal ideal is a multiplication ideal. Quasi-principal ideals have been studied in [2, 5, 9]. Let S be the set of all non-zero divisors of R and let T = {t ∈ S : tm = 0 for some m ∈ M implies m = 0}. Let RT be the localization of R at T . For any non-zero submodule N of M , let N−1 = {x ∈ RT : xN ⊆ M}. It is easily seen that N−1 is an R-submodule of RT , R ⊆ N−1 and N−1N ⊆ M . Following [10], N is an invertible submodule of M if N−1N = M . Following [10], an R module M is called a Dedekind module (Prüfer module) if every non-zero (finitely generated) submodule of M is invertible. Dedekind modules and Prüfer modules have been extensively studied in [1] and [10]. In [1, Theorem 2.3], it is proved that if R is an integral domain and M is a faithful multipli- cation R-module, then M is a Prüfer module if and only if every finitely generated submodule of M is principal. In this paper we prove that if M is a non zero faithful multiplication R- module, then R is a Prüfer module if and only if R is an integral domain and every finitely generated submodule of M is join-quasi-cyclic (i.e., join principal). Next we show that if M is a non zero faithful multiplication R-module, then R is a Dedekind module if and only if R is an integral domain and every submodule of M is a finitely generated join-quasi-cyclic submodule of M . For general background and terminology, the reader is referred to [8]. 2. Prüfer Modules and Dedekind Modules. In this paper we establish some new characterizations for Prüfer modules and Dedekind modules. We shall begin with the following lemmas. Lemma 1. Suppose M is a non zero faithful finitely generated weak-join-quasi-cyclic R-module and B is an ideal of R. If BM is weak-join-quasi-cyclic (join-quasi-cyclic), then B is weak join principal (join principal). Proof. Let A ∈ L(R). Since M is faithful and weak-join-quasi-cyclic, we have (AB : B) = (ABM : BM). As BM is weak-join-quasi-cyclic, we have (ABM : BM) = A+(0 : BM) = A+((0 : M) : B) = A+ (0 : B). Therefore B is weak join principal. Let A, C ∈ L(R). Since M is faithful and weak-join-quasi-cyclic, it follows that ((AB+ C) : B) = ((ABM + C M) : BM). As BM is join-quasi-cyclic, we have ((ABM + C M) : BM) = A+ (C M : BM) = A+ (C : B) since M is faithful and weak-join-quasi-cyclic. Therefore B is join principal. Lemma 2. Suppose M is a non zero faithful finitely generated weak-join-quasi-cyclic R-module. Suppose R is an integral domain and B is a finitely generated ideal of R. If BM is weak-join- quasi-cyclic, then B is quasi-principal. Proof. By lemma 1, B is weak join principal, so by [2, Theorem 4], B is quasi-principal. C. Jayaram and V. Raju / Eur. J. Pure Appl. Math, 3 (2010), 899-902 901 Lemma 3. Suppose R is an integral domain and M is a non zero faithful finitely generated R- module. If every finitely generated submodule of M is weak-join-quasi-cyclic, then R is a Prüfer domain. Proof. Let I be a finitely generated ideal of R. Then IM is finitely generated, so IM is weak-join-quasi-cyclic. By Lemma 2, I is quasi-principal and hence R is a Prüfer domain [8, Page 147, Ex. 10(e)]. Lemma 4. Suppose R is an arithmetical ring and M is a non zero finitely generated R-module. Then every finitely generated submodule of M is join-quasi-cyclic. Proof. Let N be a finitely generated submodule of M . It is enough to show that N is locally join-quasi-cyclic. Assume that R is a valuation ring(i.e., any two ideals are comparable). Let A ∈ L(R) and B ∈ L(M). Clearly, A+ (B : N) ⊆ ((AN + B) : N). Let a ∈ ((AN + B) : N). Then aN ⊆ AN + B. We have either (a) ⊆ A or A⊆ (a). If (a) ⊆ A, then we are through. Suppose A⊂ (a). As (a) is a multiplication ideal, it follows that A= J(a) for some proper ideal J of R. So aN ⊆ J(a)N + B, so by Nakayama’s lemma aN ⊆ B and hence a ∈ (B : N). Therefore N is join-quasi-cyclic. Lemma 5. Suppose R is an integral domain and M is a non zero faithful finitely generated R- module. Then R is a Prüfer domain if and only if every finitely generated submodule of M is join-quasi-cyclic. Proof. The proof of the lemma follows from Lemma 3 and Lemma 4. Theorem 1. Suppose M is a non zero faithful multiplication R-module. Then M is a Prüfer module if and only if R is an integral domain and every finitely generated submodule of M is join-quasi-cyclic. Proof. Suppose M is a Prüfer module. Then by [10, Theorem 3.6], R is a Prüfer domain. As R is an integral domain and M is a non zero faithful multiplication R-module, by [6, Propo- sition 3.4], M is finitely generated. Again by Lemma 5, every finitely generated submodule of M is join-quasi-cyclic. The converse part follows from [6, Proposition 3.4], Lemma 5 and [10, Theorem 3.6]. Theorem 2. Suppose M is a non zero faithful multiplication R-module. 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