On the structure of nearly epsilon and epsilon-strongly graded rings

TL;DR

This paper explores the structure of epsilon and nearly epsilon-strongly graded rings, extending Dade's theorem and introducing new categorical characterizations to better understand their composition and properties.

Contribution

It extends Dade's theorem to nearly epsilon-strongly graded rings and introduces the category SIM$S-$gr for characterizing strongly graded rings.

Findings

Extended Dade's theorem to nearly epsilon-strongly graded rings

Introduced the category SIM$S-$gr for symmetric graded modules

Provided conditions for decomposing epsilon-strongly graded rings

Abstract

In this work we study the classes of epsilon and nearly epsilon-strongly graded rings by a group $G$. In particular, we extend Dade's theorem to the realm of nearly epsilon-strongly graded rings. Moreover, we introduce the category SIM$S-$gr of symmetrically graded modules and use it to present a new characterization of strongly graded rings. A functorial approach is used to obtain a characterization of epsilon-strongly graded rings. Finally, we determine conditions for which an epsilon-strongly graded ring can be written as a direct sum of strongly graded rings and a trivially graded ring.