On generalizations of graded multiplication modules

Let $G$ be a group with identity $e$, $R$ be a $G$-graded ring with unity $1$ and $M$ be a $G$-graded $R$-module. In this article, we introduce the concept of graded quasi multiplication modules, where graded $M$ is said to be graded quasi multiplication if for every graded weakly prime $R$-submodule $N$ of $M$, $N=IM$ for some graded ideal $I$ of $R$. Also, we introduce the concept of graded absorbing multiplication modules; $M$ is said to be graded absorbing multiplication if $M$ has no graded $2$-absorbing $R$-submodules or for every graded $2$-absorbing $R$-submodule $N$ of $M$, $N=IM$ for some graded ideal $I$ of $R$. We prove many results concerning graded weakly prime submodules and graded $2$-absorbing submodules that will be useful in providing several properties of the two classes of graded quasi multiplication modules and graded absorbing multiplication modules.