On graded $J_{gr}$-classical prime submodules

Abstract

Let $G$ be a group with identity $e$. Let $R$ be a $G$-graded commutative ring with identity 1 and $M$ a graded $R$-module. A proper graded submodule $C$ of $M$ is called a graded classical prime submodule if whenever $r,sin h(R)$ and $min h(M)$ with $rsmin C$, then either $rmin C$ or $smin C$. In this paper, we introduce the concept of graded $J_{gr}$-classical prime submodule as a new generalization of graded classical submodule and we give some results concerning such graded modules. We say that a proper graded submodule $N$ of $M$ is textit{a graded }$J_{gr}$textit{-classical prime submodule of }$M$ if whenever $rsmin N$ where $r,sin h(R)$ and $min h(M)$, then either $rmin N+J_{gr}(M)$ or $smin N+J_{gr}(M)$, where $J_{gr}(M)$ is the graded Jacobson radical.