On graded weakly $J_{gr}$-semiprime submodules

Abstract

Let $\Gamma$ be a group, $\Re$ be a $\Gamma$-graded commutative ring with unity $1$ and $\Im$ a graded $\Re$-module. In this paper, we introduce the concept of graded weakly $J_{gr}$-semiprime submodules as a generalization of graded weakly semiprime submodules. We study several results concerning of graded weakly $J_{gr}$% -semiprime submodules. For example, we give a characterization of graded weakly $J_{gr}$-semiprime submodules. Also, we find some relations between graded weakly $J_{gr}$-semiprime submodules and graded weakly semiprime submodules. In addition, the necessary and sufficient condition for graded submodules to be graded weakly $J_{gr}$-semiprime submodules are investigated. A proper graded submodule $U$ of $\Im$ is said to be a graded weakly $J_{gr}$-semiprime submodule of $\Im$ if whenever $r_{g}\in h(\Re),$ $m_{h}\in h(\Im)$ and $n\in %TCIMACRO{\U{2124} }% %BeginExpansion \mathbb{Z} %EndExpansion ^{+}$ with $0\neq r_{g}^{n}m_{h}\in U$, then $r_{g}m_{h}\in U+J_{gr}(\Im)$, where $J_{gr}(\Im)$ is the graded Jacobson radical of $\Im.$