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 a multiplicatively closed subset of a commutative ring$\ R$, and let M be a unital R-module. A submodule $P\ $of $M\ $with (P:RM)∩ S=∅ is called an S% -prime submodule if there is an s∈ S\ such that am∈ P implies % sa∈(P:RM) sm∈ P., an R-module $M\ $is called S% -torsion-free if ann(M)∩ S=∅ and there exists s∈ S that $am=0\ $implies $sa=0\ $or $sm=0\ $for each a∈ R m∈ M. 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.