On graded WAG2-absorbing submodule

Abstract

Let $G$ be a group with identity $e$. Let $R$ be a $G$-graded commutative ring and $M$ a graded $R$-module. In this paper, we introduce the concept of graded $WAG2$-absorbing submodule. A number of results concerning of these classes of graded submodules and their homogeneous components are given.

Let $N=\bigoplus _{h\in G}N_{h}$ be a graded submodule of $M$ and $h\in G.$ We say that $N_{h}$ is a $h$-$WAG2$-absorbing submodule of the $R_{e}$-module $M_{h}$ if $N_{h}\neq M_{h}$; and whenever $r_{e},s_{e}\in R_{e}$ and $m_{h}\in M_{h}$ with $0\neq r_{e}s_{e}m_{h}\in N_{h}$, then either $%
r_{e}^{i}m_{h}\in N_{h}$ or $s_{e}^{j}m_{h}\in N_{h}$ or $%(r_{e}s_{e})^{k}\in (N_{h}:_{R_{e}}M_{h})$ for some $i,$ $j,$ $k$ $\in\mathbb{N}.$ We say that $N$ is {a graded }$WAG2${-absorbing submodule of }$M$ if $N\neq M$; and whenever $r_{g},s_{h}\in h(R)$ and $%m_{\lambda }\in h(M)$ with $0\neq r_{g}s_{h}m_{\lambda }\in N$, then either $r_{g}^{i}m_{\lambda }\in N$ or $s_{h}^{j}m_{\lambda }\in N$ or $%(r_{g}s_{h})^{k}\in (N:_{R}M)$ for some $i,$ $j,$ $k$ $\in \mathbb{N}.$ In particular, the following assertions have been proved:

Let $R$ be a $G$-graded ring, $M$ a graded cyclic $R$-module with $%Gr((0:_{R}M))=0$ and $N$ a graded submodule of $M.$ If $N$ is a graded $WAG2$% {-absorbing submodule of }$M,$ then\linebreak $Gr((N:_{R}M))$ is a graded $WAG2$% -absorbing ideal of $R$ (Theorem 4).

Let $R_{1}$ and $R_{2}$ be a $G$-graded rings. Let $R=R_{1}\bigoplus R_{2}$ be a $G$-graded ring and $M=M_{1}\bigoplus M_{2}$ a graded $R$-module. Let $N_{1},$ $N_{2}$ be a proper graded submodule of $M_{1}$, $M_{2}$ respectively. If $N=N_{1}\bigoplus N_{2}$ is a graded $WAG2$-absorbing submodule of $M,$ then $N_{1}$ and $N_{2}$ are graded weakly primary submodule of $R_{1}$-module $M_{1},$ $R_{2}$-module $M_{2},$ respectively. Moreover, If $N_{2}\neq 0$ $(N_{1}\neq 0),$ then $N_{1}$ is a graded weak primary submodule of $R_{1}$-module $M_{1}$ $(N_{2}$ is a graded weak primary submodule of $R_{2}$-module  $M_{2})$ (Theorem 7).