Abstract:

Let $R\ $be a commutative ring with $1\neq0$ and $M$ be an $R$-module. Suppose that $S\subseteq R\ $is a multiplicatively closed set of $R.\ $Recently Sevim et al. in \cite{SenArTeKo} introduced the notion of an $S$-prime submodule which is a generalization of a prime submodule and used them to characterize certain classes of rings/modules such as prime submodules, simple modules, torsion free modules,\ $S$-Noetherian modules and etc. Afterwards, in \cite{AnArTeKo}, Anderson et al. defined the concepts of $S$-multiplication modules and $S$-cyclic modules which are $S$-versions of multiplication and cyclic modules and extended many results on multiplication and cyclic modules to $S$-multiplication and $S$-cyclic modules. Here, in this article, we introduce and study $S$-comultiplication modules which are the dual notion of $S$-multiplication module. We also characterize certain classes of rings/modules such as comultiplication modules, $S$-second submodules, $S$-prime ideals and $S$-cyclic modules in terms of $S$-comultiplication modules. Moreover, we prove $S$-version of the dual Nakayama's Lemma.

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