Some results on modules satisfying S-strong accr∗

The rings considered in this article are commutative with identity. Modules are assumed to be unitary. Let $R$ be a ring and let $S$ be a multiplicatively closed subset of $R$. We say that a module $M$ over $R$ satisfies$S$- strong$acc{r}^{\ast }$ if for every submodule $N$ of $M$ and for every sequence $<r_n>$ of elements of $R$, the ascending sequence of submodules $\left(N{:}_M{r}_1\right)\subseteq \left(N{:}_M{r}_1{r}_2\right)\subseteq \left(N{:}_M{r}_1{r}_2{r}_3\right)\subseteq \cdots$ is $S$-stationary. That is, there exist $k\in N$ and $s\in S$ such that $s(N{:}_M{r}_1\cdots r_n)\subseteq (N{:}_M{r}_1\cdots r_k)$ for all $n\ge k$. We say that a ring $R$ satisfies $S$- strong $acc{r}^{\ast }$ if $R$ regarded as a module over $R$ satisfies $S$-strong $acc{r}^{\ast }$. The aim of this article is to study some basic properties of rings and modules satisfying $S$-strong $acc{r}^{\ast }$.