On Properties of Graded Rings with respect to Group Homomorphisms

Abstract

Let $G$ be a group and $R$ be a $G$ -graded ring with non-zero unity. The goal of our article is reconsidering some well-known concepts on graded rings using a group homomorphism $\alpha :G⟶G$ . Next is to examine the new concepts compared to the known concepts. For example, it is known that $\left(R,G\right)$ is weak if whenever $g\in G$ such that $R_g=0$ , then $R_{g^{-1}}=0$ . In this article, we also introduce the concept of $\alpha$ -weakly graded rings, where $\left(R,G\right)$ is said to be $\alpha$ -weak whenever $g\in G$ such that $R_g=0$ , and $R_{\alpha \left(g\right)}=0$ . Note that if $G$ is abelian, then the concepts of weakly and $\alpha$ -weakly graded rings coincide with respect to the group homomorphism $\alpha \left(g\right)=g^{-1}$ . We introduce an example of non-weakly graded ring that is $\alpha$ -weak for some $\alpha$ . Similarly, we establish and examine the concepts of $\alpha$ -non-degenerate, $\alpha$ -regular, $\alpha$ -strongly, $\alpha$ -first strongly graded rings, and $\alpha$ -weakly crossed product.