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

Let $ \Gamma $ be a group, $ \mathcal{A} $ be a $ \Gamma $-graded commutative ring with unity $ 1, $ and $ \mathcal{D} $ a graded $ \mathcal{A} $-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 $ \mathcal{D} $ is said to be a graded weakly $ J_{gr} $-semiprime submodule of $ \mathcal{D} $ if whenever $ r_{g}\in h(\mathcal{A}), $ $ m_{h}\in h(\mathcal{D}) $ and $ n\in \mathbb{Z} ^{+} $ with $ 0\neq r_{g}^{n}m_{h}\in U $, then $ r_{g}m_{h}\in U+J_{gr}(\mathcal{D}) $, where $ J_{gr}(\mathcal{D}) $ is the graded Jacobson radical of $ \mathcal{D}. $