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

Let $G$ be a group with identity $e$. $R$ commutative $G$-graded ring non-zero identity, $S\subseteq h(R)$ multiplicatively closed subset of and $M$ graded $R$-module. In this article, we introduce study the concept $S$-1-absorbing prime submodules. A submodule $N$ $(N:_{R}M)\cap S=\emptyset$ is said to 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 $m_{k}\in h(M)$. Some examples, characterizations properties submodules are given. Moreover, give some in multiplicative modules.