Induced quotient group gradings of epsilon-strongly graded rings

TL;DR

This paper investigates how epsilon-strongly graded rings behave under quotient group gradings, providing conditions for when the induced grading remains epsilon-strong and illustrating cases where it does not.

Contribution

It introduces necessary and sufficient conditions for the preservation of epsilon-strong grading under quotient group gradings and provides a counterexample demonstrating when it fails.

Findings

Induced quotient group grading may not be epsilon-strong.

Necessary and sufficient conditions for epsilon-strongness preservation.

Counterexample illustrating failure of epsilon-strongness in certain cases.

Abstract

Let $G$ be a group and let $S=\bigoplus_{g \in G} S_g$ be a $G$-graded ring. Given a normal subgroup $N$ of $G$, there is a naturally induced $G/N$-grading of $S$. It is well-known that if $S$ is strongly $G$-graded, then the induced $G/N$-grading is strong for any $N$. The class of epsilon-strongly graded rings was recently introduced by Nystedt, \"Oinert and Pinedo as a generalization of unital strongly graded rings. We give an example of an epsilon-strongly graded partial skew group ring such that the induced quotient group grading is not epsilon-strong. Moreover, we give necessary and sufficient conditions for the induced $G/N$-grading of an epsilon-strongly $G$-graded ring to be epsilon-strong. Our method involves relating different types of rings equipped with local units (s-unital rings, rings with sets of local units, rings with enough idempotents) with generalized epsilon-strongly graded rings.