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 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 R g = 0 , then R g − 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 R g = 0 , and R α g = 0 . Note that if G is abelian, then the concepts of weakly and α -weakly graded rings coincide with respect to the group homomorphism α g = 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.