Bulletin of the
Korean Mathematical Society BKMS

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