On graded 1-absorbing prime submodules

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