On Prime and Semiprime Submodules of Multiplication Modules

Let R be a commutative ring. An R-module M is called a multiplication module if for each submodule N of M, N = IM, for some ideal I of R. In this paper, first we define the notion of residual of a submodule in M and then we obtain some related results. In particular, we state and prove some equivalent conditions for prime and semiprime submodules of a multiplication module.