On graded weakly Jgr-semiprime submodules.

Let Γ be a group, A be a Γ-graded commutative ring with unity 1, and D a graded 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 D is said to be a graded weakly J_gr-semiprime submodule of D if whenever r_g ∈ h(A); m_h ∈ h(D) and n ∈ Z⁺ with 0 ≠ rⁿ_gm_h ∈ U, then r_gm_h ∈ U + J_gr(D), where J_gr(D) is the graded Jacobson radical of D. [ABSTRACT FROM AUTHOR]