On $gr$-quasi-semiprime submodules

Abstract

Let $G$ be a group. A ring $R$ is called a graded ring (or $G$-graded ring) if there exist additive subgroups $R_{\alpha }$ of $R$ indexed by the elements $\alpha \in G$ such that $R=\bigoplus_{\alpha \in G}R_{\alpha }$ and $R_{\alpha }R_{\beta }\subseteq R_{\alpha \beta }$ for all $\alpha $, $% \beta \in G$. If an element of $R$ belongs to $h(R)=\cup _{\alpha \in G}R_{\alpha }$, then it is called a homogeneous. A Left $R$-module $M$ is said to be \textit{a graded }$R$\textit{-module} if there exists a family of additive subgroups $\{M_{\alpha }\}_{\alpha \in G}$ of $M$ such that $% M=\bigoplus_{\alpha \in G}M_{\alpha }$ and $R_{\alpha }M_{\beta }\subseteq M_{\alpha \beta }$ for all $\alpha ,\beta \in G.$ Also if an element of $M$ belongs to $\cup _{\alpha \in G}M_{\alpha }=h(M)$, then it is called a homogeneous. A submodule $N$ of $M$ is said to be \textit{a graded submodule of }$M$ if $N=\bigoplus_{\alpha \in G}(N\cap M_{\alpha }):=\bigoplus_{\alpha \in G}N_{\alpha }$. Let $G$ be a group with identity $e$. Let $R$ be a $G$% -graded commutative ring and $M$ a graded $R$-module. A proper graded submodule $S$ of $M$ is said to be \textit{a graded semiprime (}shortly $gr$% \textit{-semiprime) submodule} if whenever $r^{n}m\in S$ where $r\in h(R)$, $% m\in h(M)$ and $n\in Z^{+}$, then $rm\in S.$ In this work, we introduce the concept of graded quasi-semiprime (shortly $gr$-quasi-semiprime) submodule as a generalization of $gr$-semiprime submodule and give some basic properties of these classes of graded submodules. We say that a proper graded submodule $S$ of $M$ is a $gr$-quasi-semiprime submodule if $% (S:_{R}M)=\{r\in R:rM\subseteq S\}$ is a $gr$-semiprime ideal of $R$.