Graded S-1-absorbing prime submodules in graded multiplication modules

Abstract

Let $G$ be a group with identity $e$. Let $R$ be a commutative $G$-graded ring with non-zero identity, $S\subseteq h(R)$ a multiplicatively closed subset of $R$ and $M$ a graded $R$-module. In this article, we introduce and study the concept of graded $S$-1-absorbing prime submodules. A graded submodule $N$ of $M$ with $(N:_{R}M)\cap S=\emptyset$ is said to be graded $S$-1-absorbing prime, if there exists an $s_{g}\in S$ such that whenever $a_{h}b_{h'}m_{k}\in N$, then either $s_{g}a_{h}b_{h'}\in (N:_{R}M)$ or $s_{g}m_{k}\in N$ for all non-unit elements $a_{h},b_{h'}\in h(R)$ and all $m_{k}\in h(M)$. Some examples, characterizations and properties of graded $S$-1-absorbing prime submodules are given. Moreover, we give some characterizations of graded $S$-1-absorbing prime submodules in graded multiplicative modules.