A generalization of pure submodules

Document Type : Research Paper


University of Farhangian, Tehran, Iran.


‎Let $R$ be a commutative ring with identity‎, ‎$S$ a multiplicatively closed subset of $R$‎, ‎and $M$ be an $R$-module‎.
‎The goal of this work is to introduce the notion of $S$-pure submodules of $M$ as a generalization of pure submodules of $M$ and prove a number of results concerning of this class of modules‎.
‎We say that a submodule $N$ of $M$ is \emph {$S$-pure} if there exists an $s \in S$ such that $s(N \cap IM) \subseteq IN$ for every ideal $I$ of $R$‎. ‎Also‎, ‎We say that $M$ is \emph{fully $S$-pure} if every submodule of $M$ is $S$-pure‎.