IDEALS AND SUBMODULES OF MULTIPLICATION MODULES

Title & Authors
IDEALS AND SUBMODULES OF MULTIPLICATION MODULES
LEE, SANG CHEOL; KIM, SUNAH; CHUNG, SANG-CHO;

Abstract
Let R be a commutative ring with identity and let M be an R-module. Then M is called a multiplication module if for every submodule N of M there exists an ideal I of R such that N = 1M. Let M be a non-zero multiplication R-module. Then we prove the following: (1) there exists a bijection: N(M)$\small{\bigcap}$V(ann$\small{\_{R}}$(M))$\small{\rightarrow}$Spec$\small{\_{R}}$(M) and in particular, there exists a bijection: N(M)$\small{\bigcap}$Max(R)$\small{\rightarrow}$Max$\small{\_{R}}$(M), (2) N(M) $\small{\bigcap}$ V(ann$\small{\_{R}}$(M)) = Supp(M) $\small{\bigcap}$ V(ann$\small{\_{R}}$(M)), and (3) for every ideal I of R, The ideal $\small{\theta}$(M) = $\small{\sum}$$\small{\_{m}$(Rm :R M) of R has proved useful in studying multiplication modules. We generalize this ideal to prove the following result: Let R be a commutative ring with identity, P $\small{\in}$ Spec(R), and M a non-zero R-module satisfying (1) M is a finitely generated multiplication module, (2) PM is a multiplication module, and (3) P$\small{^{n}}$M$\small{\neq}$P$\small{^{n+1}}$ for every positive integer n, then $\small{\bigcap}$$\small{^{}$$\small{\_{n=1}}$(P$\small{^{n}}$ + ann$\small{\_{R}}$(M)) $\small{\in}$ V(ann$\small{\_{R}}$(M)) = Supp(M) $\small{\subseteq}$ N(M).
Keywords
prime submodules;maximal submodulesq;finitely gener­ated modules;multiplication modules;
Language
English
Cited by
1.
Modules Satisfying the S-Noetherian Property and S-ACCR, Communications in Algebra, 2016, 44, 5, 1941
2.
SOME PROPERTIES OF GR-MULTIPLICATION MODULES, Korean Journal of Mathematics, 2012, 20, 3, 315
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