Given a finite group G acting as automorphisms on a ring A, the skew group ring A∗G is an important tool for studying the structure of G-stable ideals of A. The ring A∗G is G-graded, i.e.G coacts on A∗G. The Cohen-Montgomery duality says that the smash product A ∗ G#k[G]∗ of A ∗ G with the dual group ring k[G]∗ is isomorphic to the full matrix ring Mn(A) over A, where n is the order of G. In th...

Let $G=(V,E)$ be a simple graph. A set $Ssubseteq V$ isindependent set of $G$,  if no two vertices of $S$ are adjacent.The  independence number $alpha(G)$ is the size of a maximumindependent set in the graph. In this paper we study and characterize the independent sets ofthe zero-divisor graph $Gamma(R)$ and ideal-based zero-divisor graph $Gamma_I(R)$of a commutative ring $R$.

Abstract Let $$E*G$$ E ∗ G be a crossed product of division ring E and locally indicable group G . Hughes showed that up to -isomorphism, there exists at most one Hughes-free -ring. However, the existence -ring $${\mathcal {D}}_{E*G}$$ <mml...

‎Let $G$ be a group with identity $e.$ Let $R$ be a $G$-graded‎ ‎commutative ring and $M$ a graded $R$-module‎. ‎In this paper‎, ‎we‎ ‎introduce several results concerning graded classical prime‎ ‎submodules‎. ‎For example‎, ‎we give a characterization of graded‎ ‎classical prime submodules‎. ‎Also‎, ‎the relations between graded‎ ‎classical prime and graded prime submodules of $M$ are studied‎.‎

Let $G$ be a group with identity $e$. Let $R$ be a $G$-graded commutative ring with a non-zero identity and $M$ be a graded $R$-module. In this article, we introduce the concept of graded almost semiprime submodules. Also, we investigate some basic properties of graded almost semiprime and graded weakly semiprime submodules and give some characterizations of them.

let $g$ be a group with identity $e$. let $r$ be a $g$-graded commutative ring with a non-zero identity and $m$ be a graded $r$-module. in this article, we introduce the concept of graded almost semiprime submodules. also, we investigate some basic properties of graded almost semiprime and graded weakly semiprime submodules and give some characterizations of them.

We describe the essential algebra, kBTˆ(G), of Burnside biset functor shifted by a group T, at G, in two cases. First, when G and T are both finite abelian groups k is field characteristic 0. In this case, kBTˆ(G) isomorphic to quotient star which defined terms subgroups G×G×T. The second case any satisfying (|G|,|T|)=1 commutative unitary ring. semidirect product Out(G) kBZ(G)(T), monomial rin...

Let G be a group and R G-graded ring. In this paper, we present examine the concept of graded weakly 2-absorbing ideals as in generality prime ring which is not commutative, demonstrates that symmetry obtained lot outcomes commutative rings remain are commutative.