Chain conditions for rings with enough idempotents with applications to category graded rings

TL;DR

This paper establishes criteria for when rings with sufficient idempotents are artinian or noetherian based on local properties, and applies these to category graded rings, extending previous group graded results.

Contribution

It generalizes existing results from group graded rings to category and groupoid graded rings, providing new criteria for ring properties using local idempotent conditions.

Findings

Criteria for artinian and noetherian properties based on local idempotent conditions

Extension of group graded ring results to category and groupoid graded rings

Application to skew category algebras

Abstract

We obtain criteria for when a ring with enough idempotents is left/right artinian or noetherian in terms of local criteria defined by the associated complete set of idempotents for the ring. We apply these criteria to object unital category graded rings in general and, in particular, to the class of skew category algebras. Thereby, we generalize results by Nastasescu-van Oystaeyen, Bell, Park and Zelmanov from the group graded case to groupoid, and in some cases category, gradings.