S-Maximal Prime Submodules

Abstract

In this work, we propose a novel extension of the notion of prime submodules for unital modules over commutative rings with identity, called S-maximal prime submodules. A proper submodule Nof an R-module M is called  S-maximal prime if whenever rm∈N, where r∈R and m∈M, then either m∈N+SJ(N) or rM⊆N+SJ(N), where SJ(N) denotes the S-Jacobson radical of N. This concept extends the classical notion of prime submodules. In particular, We prove that prime submodules are necessarily S-maximal prime submodules, while the converse fails in general. Moreover, if an ideal I of a ring R is an S-maximal prime submodule of the R-module M, then I is called an S-maximal prime ideal of R. Several fundamental properties and characterizations of S-maximal prime submodules are investigated. Furthermore, we introduce the notion of the S-maximal prime radical of submodules and establish some of its basic properties. Special attention is given to multiplication modules, where strong connections between S-maximal prime submodules and compressible factor modules are obtained.