On S-prime submodules

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 subset of 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) boolean AND S = empty set is called an S-prime submodule if there is an s is an element of S such that am is an element of P implies sa is an element of(P : (R) M) or sm is an element of P: Also, an R-module M is called S-torsion-free if ann(M) boolean AND S = empty set and there exists s is an element of S such that am = 0 implies sa = 0 or sm = 0 for each a is an element of R and m is an element of 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