On S-prime submodules

Authors: ESRA ŞENGELEN SEVİM, TARIK ARABACI, ÜNSAL TEKİR, SUAT KOÇ

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\subseteq 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 $(P:_{R}M)\cap S=\emptyset$ is called an $S$% -prime submodule if there is an $s\in S$\ such that $am\in P$ implies $% sa\in(P:_{R}M)\ $or $sm\in P.\ $Also, an $R$-module $M\ $is called $S$% -torsion-free if $ann(M)\cap S=\emptyset$ and there exists $s\in S\ $such that $am=0\ $implies $sa=0\ $or $sm=0\ $for each $a\in R\ $and $m\in 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.

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