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 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: ₑM) S= is called an S% -prime submodule if there is an s S\ such that am P implies % sa (P: ₑM) \ 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.