On $S$-comultiplication modules

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