On graded weakly Jgr-semiprime submodules

Let Gamma be a group, 5t be a Gamma-graded commutative ring with unity 1, and D a graded 5tmodule. In this paper, we introduce the concept of graded weakly Jgr-semiprime submodules as a generalization of graded weakly semiprime submodules. We study several results concerning of graded weakly Jgr-semiprime submodules. For example, we give a characterization of graded weakly Jgr-semiprime submodules. Also, we find some relations between graded weakly Jgr-semiprime submodules and graded weakly semiprime submodules. In addition, the necessary and sufficient condition for graded submodules to be graded weakly Jgr-semiprime submodules are investigated. A proper graded submodule U of D is said to be a graded weakly Jgr-semiprime submodule of D if whenever rg E h(5t), mh E h(D) and n E Z+ with 0 # rgnmh E U, then rgmh E U + Jgr(D), where Jgr(D) is the graded Jacobson radical of D.