Peirce decompositions, idempotents and rings

TL;DR

This paper explores how Peirce decompositions induced by idempotents can be used to classify and unify various classes of rings, expanding classical theories with new examples and applications.

Contribution

It introduces a framework that generalizes classical ring theory by using Peirce decompositions to analyze and classify broader classes of rings.

Findings

Peirce decompositions provide a unifying structure for ring classification.

The paper extends classical semiperfect ring theory to larger classes.

Applications demonstrate the utility of the new framework.

Abstract

Idempotents dominate the structure theory of rings. The Peirce decomposition induced by an idempotent provides a natural environment for defining and classifying new types of rings. This point of view offers a way to unify and to expand the classical theory of semiperfect rings and idempotents to much larger classes of rings. Examples and applications are included.