Abstract
AbstractLetGbe an abelian group with identityee. LetRbe aG-graded commutative ring with identity 1, andMMbe a gradedR-module. In this paper, we introduce the concept of gradedJgr{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 submoduleCCofMMis called a gradedJgr{J}_{gr}-classical 2-absorbing submodule ofMM, if wheneverrg,sh,ti∈h(R){r}_{g},{s}_{h},{t}_{i}\in h\left(R)andxj∈h(M){x}_{j}\in h\left(M)withrgshtixj∈C{r}_{g}{s}_{h}{t}_{i}{x}_{j}\in C, then eitherrgshxj∈C+Jgr(M){r}_{g}{s}_{h}{x}_{j}\in C+{J}_{gr}\left(M)orrgtixj∈C+Jgr(M){r}_{g}{t}_{i}{x}_{j}\in C+{J}_{gr}\left(M)orshtixj∈C+Jgr(M),{s}_{h}{t}_{i}{x}_{j}\in C+{J}_{gr}\left(M),whereJgr(M){J}_{gr}\left(M)is the graded Jacobson radical.
Highlights
Introduction and preliminariesThroughout this paper, all rings are commutative with identity and all modules are unitary
The concept of graded 2-absorbing submodules as a generalization of graded prime submodules was introduced by Al-Zoubi and Abu-Dawwas in [12] and studied in [13,14]
We study the behaviour of graded Jgr-classical 2-absorbing submodule under graded homomorphisms and under localization
Summary
Introduction and preliminariesThroughout this paper, all rings are commutative with identity and all modules are unitary. We give a characterization of graded Jgr-classical 2-absorbing submodule (see Theorem 2.10). Recall from [6] that a proper graded submodule C of a graded R-module M is said to be a graded classical 2-absorbing submodule of M if whenever rg, sh, ti ∈ h(R) and xj ∈ h(M) with rg shti xj ∈ C, either rg sh xj ∈ C or rg ti xj ∈ C or sh ti xj ∈ C.
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