On S-prime submodules

Abstract

ABSTRACT: In this study, we introduce the concepts of S -prime submodules and S -torsion-free modules, which are generalizations of prime submodules and torsion-free modules. Suppose S ? R is a multiplicatively closed subset of a commutative ring R, and let M be a unital R-module. A submodule P of M with (P :R M) ? S = ? is called an S -prime submodule if there is an s ? S such that am ? P implies sa ? (P :R M) or sm ? P. Also, an R-module M is called S -torsion-free if ann(M) ? S = ? and there exists s ? S such that am = 0 implies sa = 0 or sm = 0 for each a ? R and m ? M. In addition to giving many properties of S -prime submodules, we characterize certain prime submodules in terms of S -prime submodules. Furthermore, using these concepts, we characterize some classical modules such as simple modules, S -Noetherian modules, and torsion-free modules.