On Properties of Graded Rings with respect to Group Homomorphisms

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 α:G⟶G. Next is to examine the new concepts compared to the known concepts. For example, it is known that R,G is weak if whenever g∈G such that Rg=0, then Rg−1=0. In this article, we also introduce the concept of α-weakly graded rings, where R,G is said to be α-weak whenever g∈G such that Rg=0, and Rαg=0. Note that if G is abelian, then the concepts of weakly and α-weakly graded rings coincide with respect to the group homomorphism αg=g−1. We introduce an example of non-weakly graded ring that is α-weak for some α. Similarly, we establish and examine the concepts of α-non-degenerate, α-regular, α-strongly, α-first strongly graded rings, and α-weakly crossed product.