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Some Remarks On Multiplication Modules

P. Smith
Published 1988 · Mathematics

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In this note all rings are commutative rings with identity and all modules are unital. Let R be a commutative ring with identity. 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 = I M. Various properties of multiplication modules are considered. If there is a common theme it is that the methods used generalise results of Naoum and Hasan proved using matrix methods. 1. Sums and intersections. Let R be a commutative ring with identity and M a unital R-module. The annihilator of M is denoted ann (M) and for any m ~ M the annihilator ofm is denoted ann(m). IfN is a submodule of M then (N: M) denotes the ideal ann(M/N) of R, that is (N :M) = {r e R: rM c= N}. 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 = I M. It is clear that every cyclic R-module is a multiplication module. Let N be a submodule of a multiplication module M. There exists an ideal I of R such that N = I M. Note that I~(N:M) and N=IM=(N:M) M~N so that N=(N:M) M. It follows that an R-module M is a multiplication module if and only if N = (N : M) M for all submodules N of M. An ideal A of R which is a multiplication module is called a multiplication ideal. Let P be a maximal ideal of a ring R. An R-module M is called P-torsion provided for each m ~ M there exists p E P such that (1 - p) m = 0. On the other hand M is called P-cyclic provided there exist x ~ M and q E P such that (1 - q) M ~ R x. Our starting point is the following result taken from [3, Theorem 1.2].
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