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...

Let $G$ be a finitely 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...

For any complex valued functions over topological space F there exists a relation in von Neumann algebras of *-graded that is bounded on compact Hausdorff where for category- I, II, III commutative form 〖AW〗^* such to satisfy monotone complete C^* algebra suffice an isomorphic factor f the same tamed as W^* having generators η generic group η(G) 2-groups G_+ and G_- former being additive intege...

background: in recent years, neoadjuvant chemoradiation and subsequent surgical resection with total mesorectal excision has been shown to increase local control with decreased toxicity. neoadjuvant chemoradiotherapy is the standard treatment for locally advanced rectal cancer. in this study we evaluated the efficacy a cox-2 inhibitor on pathologic response, sphincter preservation and acute tox...

We show how to construct a graded locally compact Hausdorff \'etale groupoid from C*-algebra carrying coaction of discrete group, together with suitable abelian subalgebra. call this the extended Weyl groupoid. When is trivial and subalgebra Cartan, our agrees Renault's prove that if G second-countable grading interior trivially isotropy torsion free, then its reduced isomorphic as G. In partic...

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

Let $G$ be a group with identity $e$. $R$ $G$-graded commutative ring and $M$ graded $R$-module. In this paper, we introduce the concept of classical strongly 2-absorbing second submodules modules over rings. A number results concerning these classes their homogeneous components are given.