International Journal of Algebra, Vol. 6, 2012, no. 23, 1117 - 1120 Cyclic and Multiplication P -Be´zout Modules Muhamad Ali Misri Department of Mathematics Education, IAIN Syekh Nurjati Cirebon Jl. By Pass Perjuangan Kesambi, Cirebon, Indonesia [email protected], [email protected] Irawati Department of Mathematics, Institut Teknologi Bandung Jl. Ganesa 10 Bandung 40132, Indonesia [email protected], [email protected] Hanni Garminia Y Department of Mathematics, Institut Teknologi Bandung Jl. Ganesa 10 Bandung 40132, Indonesia [email protected] Abstract We study P -B´ezout, Noetherian and multiplication modules. We inves- tigate the sufficient condition of a module over P -B´ezout ring is to be P -B´ezout module and even for arbitrary ring. Let R be a commutative ring with an identity and M be a cyclic multiplication R-module which has some properties of its submodule, we prove that M is P -B´ezout module. Mathemaics Subject Classification: 13C05,13C99 Keywords: P -b´ezout module, finitely generated submodule, prime sub- module, cyclic submodule 1 Introduction The ring which considered in this paper is commutative with an identity and will be denoted by R. The concept of P -B´ezout ring was introduced and studied in Bakkari [1] as a generalization of the B´ezout ring. A ring R is said to be P -B´ezout if every finitely generated prime ideal I over R is principle. 1118 M. A. Misri, Irawati and H. Garminia Y We will adapt that concept into module. A module M over R is said to be P -B´ezout if every finitely generated prime submodule N of M is cyclic. See [3] for example. Let R be a commutative ring with an identity and M be a cyclic multiplication R-module which has some properties of its submodule, we prove that M is P -B´ezout module. In section 2, we will explain about a cyclic and multiplication module and we use this in section 3 to prove the main theorem. 2 Cyclic and Multiplication Module An R-module M is said to be cyclic if it has one generator, that is if M = Rm for some m ∈ M. Similarly, M is finitely generated if it has finitely generators, that is if M = Rm1 + Rm2 + Rm3 + ... + Rmn for finitely many mi ∈ M. A module M is said to be Noetherian if every submodule of M is finitely generated [5, Page 10]. Following [2], the annihilator of M is denoted Ann(M). For any submodule N of M, the annihilator of the factor module M/N will be denoted by (N : M) so that (N : M)={r ∈ R|rM ⊆ N}. We can see that (N : M) is an ideal of R. A module M is said to be faithful if Ann(M)=0}. A module M is said to be multiplication module if for any submodule N of M, there exist an ideal I of R such that N = IM. It was proven that M is a multiplication module if and only if N =(N : M)M for every submodule N of M. A proper submodule N of M over R is said to be prime if rm ∈ N for r ∈ R and m ∈ M implies that either m ∈ N or r ∈ (N : M). It clear that N is prime submodule of M if IK ⊆ N for submodule K of M and ideal I of R implies that either K ⊆ N or I ⊆ (N : M). Some properties which used in later section is given as follow. Let M be a faithful multiplication R-module. Lee [4, Theorem 2.6] has proved that M is finitely generated. Let M be a multiplication R-module, N be a submodule of M and let I be an ideal of R contained in the Jacobson radical of R. Ghalanzarzadeh at all [6, Corollary 3.10] said that if IM + N = M then N = M. Cyclic and multiplication P -Be´zout modules 1119 3 Properties of P-Be´zout Module In this section we’ll prove properties of P -B´ezout modules in two theorems. The first theorem is for a P -B´ezout ring and the second is for arbitrary ring. We need the following lemma to prove the first theorem. Lemma 3.1 Let R be a commutative ring, M be a module over R and N be a prime submodule of M. Then ideal (N : M) of R is prime. Proof. Suppose AB ⊆ (N : M) for A and B are ideals of R. By definition of annihilator (N : M), it yields (N : M)M ⊆ N and hence ABM ⊆ N since ABM ⊆ (N : M)M. Furthermore, by N is a prime submodule then A ⊆ (N : M)orB ⊆ (N : M) since BM ⊆ N. Since A ⊆ (N : M)or B ⊆ (N : M) so that (N : M) is prime ideal has proved. 2 The first theorem, we mean, as follows: Theorem 3.2 Let R be a Noetherian P -Be´zout ring and M be a faithful multiplication module. Then every prime submodules of M are finitely gener- ated and of the form rM for <r>is a prime ideal of R. Furthermore, if either M is a cyclic or simple module then M is P -Be´zout. Proof. Let N be a prime submodule of M. By Lemma 3.1, (N : M) is prime ideal of R. Since R is a Noetherian ring then (N : M) is finitely generated ideal and hence (N : M) is principle by R is a P -B´ezout. Let (N : M)=Rr for r ∈ R.SoN =(N : M)M = rM since M is a multiplica- tion module. Since M is a faithful multiplication module hence N is finitely generated by [4, Theorem 2.6]. Therefore N is finitely generated and of the form rM for <r>is a prime ideal of R. Let m ∈ M so that N = rM = rRm = Rrm since M is a cyclic module and hence N is cyclic submodule since N = Rx for x = rm ∈ M. Since every finitely generated prime submodules of M is cyclic then M is a P -B´ezout module. Pure submodule of M is only 0 Since M is a simple module. If N =0 clear that N is cyclic finitely generated prime submmodule and hence M is a P -B´ezout module. 2 Finally, we come to the second properties on this study of P -B´ezout module as follow. 1120 M. A. Misri, Irawati and H. Garminia Y Theorem 3.3 Let R be a commutative ring and M be a cyclic multiplica- tion module. If every finitely generated prime submodules of M are maximal and for every maximal submodules N of M, there exist ideal I of R such that IM ⊆ N with I ⊆ Rad(R). Then M is a P -Be´zout. Proof. Let N be a finitely generated prime submodule of M. By hipothesis of theorem yields that N is a maximal submodule and the same way, there exist ideal I of R such that IM ⊆ N with I ⊆ Rad(R). So N + IM = M and hence N = M by [6, Corollary 3.10]. Therefore N is cyclic submodule since M is a cyclic module. Since every finitely generated prime submodules of M is cyclic then M is a P -B´ezout module. 2 References [1] Chahrazade Bakkari, On P-b´ezout rings, International Journal of Algebra, 3 (2009), 669 - 673. [2] A. Barnard, Multiplication modules, Journal of Algebra, 71 (1981), 174 - 178. [3] Muhamad Ali Misri, Irawati and Hanni Garminia Y., Generalization of b´ezout module, Far East Journal of Mathematical Sciences (FJMS), Ac- cepted. [4] Dong-Soo Lee and Hyun-Bok Lee, Some remarks on faithful multiplication modules, Journal of The Chungcheong Mathematical Society, 6 (1993), 131 - 137. [5] Donald S. Passman, A Course in Ring Theory, Wadsworth & Brooks/Cole Advanced Books & Software, USA, 1991. [6] Sh. Ghalanzarzadeh, P Malakoti Rad and S. Shirinkam, Multiplication modules and Cohen theorem, Mathematical Sciences, 2 (2008), 251-260. Received: June, 2012.