S-Semiprime Submodules and S-Reduced Modules

Abstract EN

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∈S, and whenever rnm∈P for some r∈R,m∈M, and n∈ℕ, then srm∈P.

Also, M is said to be an S-reduced module if there exists (fixed) s∈S, and whenever rnm=0 for some r∈R,m∈M, and n∈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.