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 2 h(R) and m 2 h(M) with rsm 2 C, then either rm 2 C or sm 2 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 a Graded Jgr-classical prime submodule of M if whenever rsm 2 N where r,s 2 h(R) and m 2 h(M), then either rm 2 N + Jgr(M) or sm 2 N + Jgr(M), where Jgr(M) is the Graded Jacobson radical.

Let $R= \bigoplus_{g \in G} R_g$ be a $G-$Graded commutative ring with identity, $I$ be a Graded ideal and let $M$ a $G-$Graded unitary $R$-module, where $G$ is a semigroup with identity $e$. We introduce Graded $I-$prime ideals (submodules) as a generalizations of the classical notions of prime ideals (submodules). We show that the new notions inherite the basic properties of the classical ones. In particular, we investigate the localization theory of these two concepts. We prove that for a faithfull flat module $F$, a Graded submodule $P$ of $M$ is $I-$prime if and only if $F \otimes P$ is Graded $I-$prime submodule of $F \otimes M$. As an application, for finitely generated Graded module $M$ over Noetherian Graded ring $R$, the completion of Graded $I-$prime submodules is $I-$prime submodule.

Let R be a Graded ring and M be a Graded R -module. We define a topology on Graded prime spectrum G- Spec(M) of the Graded R -module M which is analogous to that for G- Spec(R), and investigate several properties of the topology.

Let G be a , nitely generated abelian group and M be a G-Graded A-module. In general, G-associated prime ideals to M may not exist. In this paper, we introduce the concept of G-attached prime ideals to M as a generalization of G-associated prime ideals which gives a connection between certain G-prime ideals and G-Graded modules over a (not necessarily G-Graded Noetherian) ring. We prove that the G-attached prime ideals exist for every nonzero G-Graded module and this generalization is proper. We transfer many results of G-associated prime ideals to G-attached prime ideals and give some applications of it.

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 R be a G-Graded ring and M be a G-gr-R-module. In this article, we introduce the concept of Graded weakly classical prime submodules and give some properties of such a submodule.

Let G be a group. Let R be a G-Graded commutative ring with identity and let M be a Graded R-module. The Graded classical prime spectrum Cl: Specg(M) is de, ned to be the set of all Graded classical prime submodule of M. In this paper we estab-lish necessary and su, cient conditions for Cl: Specg(M) with the Zariski topology to be a Noetherian topological space.