On graded Jgr -classical 2-absorbing submodules of graded modules over graded commutative rings

Let G be an abelian group with identity e e . Let R be a G-graded commutative ring with identity 1, and M M be a graded R-module. In this paper, we introduce the concept of graded J g r {J}_{gr} -classical 2-absorbing submodule as a generalization of a graded classical 2-absorbing submodule. We give some results concerning of these classes of graded submodules. A proper graded submodule C C of M M is called a graded J g r {J}_{gr} -classical 2-absorbing submodule of M M , if whenever r g , s h , t i ∈ h ( R ) {r}_{g},{s}_{h},{t}_{i}\in h\left(R) and x j ∈ h ( M ) {x}_{j}\in h\left(M) with r g s h t i x j ∈ C {r}_{g}{s}_{h}{t}_{i}{x}_{j}\in C , then either r g s h x j ∈ C + J g r ( M ) {r}_{g}{s}_{h}{x}_{j}\in C+{J}_{gr}\left(M) or r g t i x j ∈ C + J g r ( M ) {r}_{g}{t}_{i}{x}_{j}\in C+{J}_{gr}\left(M) or s h t i x j ∈ C + J g r ( M ) , {s}_{h}{t}_{i}{x}_{j}\in C+{J}_{gr}\left(M), where J g r ( M ) {J}_{gr}\left(M) is the graded Jacobson radical.