MULTIPLICATION MODULES WHOSE ENDOMORPHISM RINGS ARE INTEGRAL DOMAINS

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 GM⁄(M,N) = {f 2 M⁄ | f(M) µ N} and the other is of the form GM⁄(N,0) = {f 2 M⁄ | f(N) = 0} for a submodule N of M.