Locally finite profinite rings

TL;DR

This paper classifies the structure of locally finite profinite rings, showing they are products of matrix rings over finite fields and characterizing their radicals, with applications to small compact G-rings.

Contribution

It provides a complete classification of locally finite profinite rings and describes their radicals, extending understanding of their structure and actions in the context of small compact G-rings.

Findings

Locally finite profinite rings are products of matrix rings over finite fields.

The Jacobson radical of such rings is nilpotent with finite nil-exponent.

Results apply to actions of G on these rings in small compact G-ring contexts.

Abstract

We investigate the structure of locally finite profinite rings. We classify (Jacobson-) semisimple locally finite profinite rings as products of complete matrix rings of bounded cardinality over finite fields, and we prove that the Jacobson radical of any locally finite profinite ring is nil of finite nilexponent. Our results apply to the context of small compact $G$-rings, where we also obtain a description of possible actions of $G$ on the underlying ring.