$S$-$M$-cyclic submodules and some applications

In this paper, we introduce the notion of $S$-$M$-cyclic submodules, which is a generalization of the notion of $M$-cyclic submodules. Let $M, N$ be right $R$-modules and $S$ be a multiplicatively closed subset of a ring $R$. A submodule $A$ of $N$ is said to be an $S$-$M$-cyclic submodule, if there exist $s\in S$ and $f \in Hom_R(M,N)$ such that $As \subseteq f(M) \subseteq A$. Besides giving many properties of $S$-$M$-cyclic submodules, we generalize some results on $M$-cyclic submodules to $S$-$M$-cyclic submodules. Furthermore, we generalize some properties of principally injective modules and pseudo-principally injective modules to $S$-principally injective modules and $S$-pseudo-principally injective modules, respectively. We study the transfer of this notion to various contexts of these modules.