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 $s\in S$ , and whenever $r^{n}m\in P$ for some $r\in R,m\in M$ , and $n\in \mathbb{N}$ , then $\text{srm}\in P$ . Also, $M$ is said to be an $S$ -reduced module if there exists (fixed) $s\in S$ , and whenever $r^{n}m=0$ for some $r\in R,m\in M$ , and $n\in \mathbb{N},$ then $\text{srm}=0$ . 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.