On graded WAG2-absorbing submodule

Let $G$ be a group with identity $e$. $R$ $G$-graded commutative ring and $M$ graded $R$-module. In this paper, we introduce the concept of $WAG2$-absorbing submodule. A number results concerning these classes submodules their homogeneous components are given.
 $N=\bigoplus _{h\in G}N_{h}$ submodule $h\in G.$ We say that $N_{h}$ is $h$-$WAG2$-absorbing $R_{e}$-module $M_{h}$ if $N_{h}\neq M_{h}$; whenever $r_{e},s_{e}\in R_{e}$ $m_{h}\in M_{h}$ $0\neq r_{e}s_{e}m_{h}\in N_{h}$, then either $%r_{e}^{i}m_{h}\in N_{h}$ or $s_{e}^{j}m_{h}\in $%(r_{e}s_{e})^{k}\in (N_{h}:_{R_{e}}M_{h})$ for some $i,$ $j,$ $k$ $\in\mathbb{N}.$ $N$ {a }$WAG2${-absorbing }$M$ $N\neq M$; $r_{g},s_{h}\in h(R)$ $%m_{\lambda }\in h(M)$ r_{g}s_{h}m_{\lambda N$, $r_{g}^{i}m_{\lambda N$ $s_{h}^{j}m_{\lambda $%(r_{g}s_{h})^{k}\in (N:_{R}M)$ $\in \mathbb{N}.$ particular, following assertions have been proved:
 ring, cyclic $R$-module $%Gr((0:_{R}M))=0$ $M.$ If $WAG2$% {-absorbing }$M,$ then\linebreak $Gr((N:_{R}M))$ -absorbing ideal (Theorem 4).Let $R_{1}$ $R_{2}$ rings. $R=R_{1}\bigoplus R_{2}$ $M=M_{1}\bigoplus M_{2}$ $N_{1},$ $N_{2}$ proper $M_{1}$, $M_{2}$ respectively. $N=N_{1}\bigoplus N_{2}$ $M,$ $N_{1}$ weakly primary $R_{1}$-module $M_{1},$ $R_{2}$-module $M_{2},$ Moreover, $N_{2}\neq 0$ $(N_{1}\neq 0),$ weak $M_{1}$ $(N_{2}$ $M_{2})$ 7).