Groupoid Graded Semisimple Rings

TL;DR

This paper develops the theory of groupoid graded semisimple rings, extending classical results like Wedderburn-Artin and Jacobson-Chevalley theorems to a broader class of non-unital, locally artinian rings with groupoid grading.

Contribution

It introduces the concept of $ ext{Gamma}_0$-artinian rings, characterizes groupoid graded semisimple rings, and explores their structure, modules, and connections to semisimple categories, generalizing key algebraic theorems.

Findings

Characterization of groupoid graded semisimple rings as direct sums of graded simple $ ext{Gamma}_0$-artinian rings.

Proving a groupoid graded version of the Jacobson-Chevalley density theorem.

Identification of conditions under which all graded modules over a groupoid graded division ring are free.

Abstract

We develop the theory of groupoid graded semisimple rings. Our rings are neither unital nor one-sided artinian. Instead, they exhibit a strong version of having local units and being locally artinian, and we call them $\Gamma_0$-artinian. One of our main results is a groupoid graded version of the Wedderburn-Artin Theorem, where we characterize groupoid graded semisimple rings as direct sums of graded simple $\Gamma_0$-artinian rings and we exhibit the structure of this latter class of rings. In this direction, we also prove a groupoid graded version of Jacobson-Chevalley density theorem. We need to define and study properties of groupoid gradings on matrix rings (possibly of infinite size) over groupoid graded rings, and specially over groupoid graded division rings. Because of that, we study groupoid graded division rings and their graded modules. We consider a natural notion of freeness for groupoid graded modules that, when specialized to group graded rings, gives the usual one, and show that for a groupoid graded division ring all graded modules are free (in this sense). Contrary to the group graded case, there are groupoid graded rings for which all graded modules are free according to our definition, but they are not graded division rings. We exhibit an easy example of this kind of rings and characterize such class among groupoid graded semisimple rings. We also relate groupoid graded semisimple rings with the notion of semisimple category defined by B. Mitchell. For that, we show the link between functors from a preadditive category to abelian groups and graded modules over the groupoid graded ring associated to this category, generalizing a result of P. Gabriel. We characterize simple artinian categories and categories for which every functor from them to abelian groups is free in the sense of B. Mitchell.