Multiplication modules whose endomorphism rings are integral domains
Bull. Korean Math. Soc. 2010 Vol. 47, No. 5, 1053-1066
https://doi.org/10.4134/BKMS.2010.47.5.1053
Published online September 1, 2010
Sang Cheol Lee
Chonbuk National University
Abstract : In this paper, several properties of endomorphism rings of modules are investigated. A multiplication module $M$ over a commutative ring $R$ induces a commutative ring $M^*$ of endomorphisms of $M$ and hence the relation between the prime (maximal) submodules of $M$ and the prime (maximal) ideals of $M^*$ can be found. In particular, two classes of ideals of $M^*$ are discussed in this paper: one is of the form $G_{M^*}(M, N) = \lbrace f \in M^* \phantom{.} \vert \phantom{.} f(M) \subseteq N \rbrace$ and the other is of the form $G_{M^*}(N, 0) =\lbrace f \in M^* \phantom{.} \vert \phantom{.} f(N) =0 \rbrace$ for a submodule $N$ of $M$.
Keywords : multiplication module, semi-injective module, self-cogenerated module, tight closed submodule and closed submodule
MSC numbers : 13C05, 13C10, 13C11
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