S-Semiprime Submodules and S-Reduced Modules

Abstract

This article introduces the concept of S-semiprime submodules which are a generalization of semiprime submodules and S-prime submodules. Let M be a nonzero unital R-module, where R is a commutative ring with a nonzero identity. Suppose that S is a multiplicatively closed subset of R. A submodule P of M is said to be an S-semiprime submodule if there exists a fixed s is an element of S, and whenever rnm is an element of P for some r is an element of R,m is an element of M, and n is an element of N, then srm is an element of P. Also, M is said to be an S-reduced module if there exists (fixed) s is an element of S, and whenever rnm=0 for some r is an element of R,m is an element of M, and n is an element of N, then srm=0. In addition, to give many examples and characterizations of S-semiprime submodules and S-reduced modules, we characterize a certain class of semiprime submodules and reduced modules in terms of these concepts.