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

Abstract

Let G be an abelian group with identity ee. Let R be a G-graded commutative ring with identity 1, and MM be a graded R-module. In this paper, we introduce the concept of graded Jgr{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 CC of MM is called a graded Jgr{J}_{gr}-classical 2-absorbing submodule of MM, if whenever rg,sh,ti∈h(R){r}_{g},{s}_{h},{t}_{i}\in h\left(R) and xj∈h(M){x}_{j}\in h\left(M) with rgshtixj∈C{r}_{g}{s}_{h}{t}_{i}{x}_{j}\in C, then either rgshxj∈C+Jgr(M){r}_{g}{s}_{h}{x}_{j}\in C+{J}_{gr}\left(M) or rgtixj∈C+Jgr(M){r}_{g}{t}_{i}{x}_{j}\in C+{J}_{gr}\left(M) or shtixj∈C+Jgr(M),{s}_{h}{t}_{i}{x}_{j}\in C+{J}_{gr}\left(M), where Jgr(M){J}_{gr}\left(M) is the graded Jacobson radical.

Keywords

• graded jgr -classical 2-absorbing submodule
• graded classical 2-absorbing submodule
• graded 2-absorbing submodule
• 13a02
• 16w50