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Let R be a commutative ring and M an R-module. Then M is a multiplication module if N N : MM for each submodule N of M. The ideal yM mPM Rm : M of R has proved useful in studying multiplication modules. We show that if M is a faithful multiplication module, then yM fI an ideal of R j IM Mg tM, the trace ideal of M. Moreover, yM is an idempotent multiplication ideal of R and yyM yM. We also show that for a multiplication module M, yM=0 : M is an ideal of the endomorphism ring End R M of M and that End R M % lim 2 R=0 : N where the inverse limit is taken over the ®nitely generated submodules N of M. 3383

Abstract. Let R be a commutative ring with identity and M be a unital R-module. Then M is called a multiplication module provided for every submodule N of M there exists an ideal I of R such that N = IM. Our objective is to investigate properties of prime and semiprime submodules of multiplication modules. Mathematics Subject Classification: 13C05, 13C13Keywords: Multiplication modules, Prime submodules, Semiprime submodules Throughout this paper all rings will be commutative with identity and all modules will be unitary. Let R be a ring and M be a unital R-module. An R-module M is called a multiplication module provided for each submodule N of M there exists an ideal I of R such that N = IM. We say that I is a presantation ideal of N. Clearly, every submodule of M has a presantation ideal if and only if M is a multiplication module. Let N and K be submodules of a multiplication with N = I 1 M and K = I 2 M for some ideals I 1 and I 2 of R.The product N and K denoted by NK is defined by NK = I 1 I 2 M. Then by [6,theorem 3.4], the product of N and K is independent of presentation of N and K. Note that this definition is different from the definition of ordinary ideal multiplication. Indeed, let R = Z be the ring of integers, and let M = 2Z and N = K = 4Z. Then NK is 16Z by the usual definition and is 8Z by the our definition. Moreover, for a, b ∈ M, by ab we mean the product

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