graded classical prime submodule

Let G be a group, R be a G-graded commutative ring with identity, M be a unitary graded R-module, Specg(R) be the set of graded prime ideals of R, and Cl.Specg(M) be the set of all graded classical prime submodules of M. In this paper among other things, the author studied the Zariski topology on both and Cl.Specg(M), and investigate some properties of the Zariski topology on Cl.Specg(M) and some conditions under which the graded classical prime spectrum of M is a spectral for its Zariski topology.

Let G be a group with identity e. Let R be a G-graded commutative ring and let M be a graded R-module. The graded classical prime spectrum Cl.Specg(M) is defined to be the set of all graded classical prime submodule of M. The Zariski topology on Cl.Specg(M); denoted by ϱg. In this paper we establish necessary and sufficient conditions for Cl.Specg(M) with the Zariski topology to be a Noetherian topological space.

Let G be a group with identity e. Let R be a G-graded commutative ring with identity 1 and M a graded R-module. A proper graded submodule C of M is called a graded classical prime submodule if whenever r,s∈h(R) and m∈h(M) with rsm∈C, then either rm∈C or sm∈C. In this paper, we introduce the concept of graded Jgr-classical prime submodule as a new generalization of graded classical submodule and we give some results concerning such graded modules. We say that a proper graded submodule N of M is \textit{a graded }Jgr\textit{-classical prime submodule of \ }M if whenever rsm∈N where r,s∈h(R) and m∈h(M), then either rm∈N+Jgr(M) or sm∈N+Jgr(M), where Jgr(M) is the graded Jacobson radical.