On Graded $1$-Absorbing Prime Submodules

Abstract

Let $G$ be a group with identity $e$, $R$ be a commutative $G$-graded ring with unity $1$ and $M$ be a $G$-graded unital $R$-module. In this article, we introduce the concept of graded $1$-absorbing prime submodule. A proper graded $R$-submodule $N$ of $M$ is said to be a graded $1$-absorbing prime $R$-submodule of $M$ if for all non-unit homogeneous elements $x, y$ of $R$ and homogeneous element $m$ of $M$ with $xym\in N$, either $xy\in (N :_{R} M)$ or $m\in N$. We show that the new concept is a generalization of graded prime submodules at the same time it is a special graded $2$-absorbing submodule. Several properties of a graded $1$-absorbing prime submodule have been obtained. We investigate graded $1$-absorbing prime submodules when the components $\left\{M_{g}:g\in G\right\}$ are multiplication $R_{e}$-modules.