On Graded Semi $J_{gr}$-2-absorbing and Graded Weakly Semi $J_{gr}$-2-absorbing Submodules

Let G be a group with identity e.Let R be a G-graded commutative ring and M a graded R-module.In this paper, we introduce the concepts of graded semi Jgr-2-absorbing and graded weakly semi Jgr-2-absorbing submodules of M and study the behavior of these notions under several constructions.A proper graded submodule N of M is said to be a graded semi Jgr-2-absorbing (resp.graded weakly semi Jgr-2absorbing) submodule of M if whenever rg h(R) and m h h(M ) with r 2 g m h N (resp.0 = r 2 g m h N ), then either rgm h N + Jgr(M ) or r 2 g (N + Jgr(M ) : R M ), where Jgr(M ) is the graded Jacobson radical.