Epsilon-strongly graded rings, separability and semisimplicity

TL;DR

This paper introduces epsilon-strongly graded rings, a new class that generalizes strongly graded rings and unital partial crossed products, providing criteria for separability and illustrating their structure with examples.

Contribution

It defines epsilon-strongly graded rings, extends existing results on separability, and connects unital partial crossed products to this new class, offering new insights and proofs.

Findings

Characterization of when epsilon-strongly graded rings are separable

Generalization of known results for strongly graded rings and partial crossed products

Examples of separable epsilon-strongly graded rings with finite and infinite groups

Abstract

We introduce the class of epsilon-strongly graded rings and show that it properly contains both the class of strongly graded rings and the class of unital partial crossed products. We determine precisely when an epsilon-strongly graded ring is separable over its principal component. Thereby, we simultaneously generalize a result for strongly group graded rings by Nastasescu, Van den Bergh and Van Oystaeyen, and a result for unital partial crossed products by Bagio, Lazzarin and Paques. We also show that the class of unital partial crossed products appear in the class of epsilon-strongly graded rings in a fashion similar to how the classical crossed products present themselves in the class of strongly graded rings. Thereby, we obtain, in the special case of unital partial crossed products, a short proof of a general result by Dokuchaev, Exel and Sim\'{o}n concerning when graded rings can be presented as partial crossed products. We also provide some interesting classes of examples of separable epsilon-strongly graded rings, with finite as well as infinite grading groups. In particular, we obtain an answer to a question raised by Le Bruyn, Van den Bergh and Van Oystaeyen in 1988.