Commutators and Anti-Commutators of Idempotents in Rings

TL;DR

This paper characterizes rings with idempotents whose commutator or anti-commutator is invertible, especially focusing on matrix rings, and provides new criteria for identifying such rings and their properties.

Contribution

It establishes a characterization of rings with invertible commutators or anti-commutators of idempotents, especially in matrix rings, and introduces new criteria for recognizing 2x2 matrix rings.

Findings

Rings with invertible commutators of idempotents are isomorphic to matrix rings over rings with sum of two units.

In such rings, the anti-commutator of idempotents is automatically invertible.

A ring is a 2x2 matrix ring if and only if it has an invertible commutator of an idempotent and an element.

Abstract

We show that a ring $\,R\,$ has two idempotents $\,e,e'\,$ with an invertible commutator $\,ee'-e'e\,$ if and only if $\,R \cong {\mathbb M}_2(S)\,$ for a ring $\,S\,$ in which $\,1\,$ is a sum of two units. In this case, the "anti-commutator" $\,ee'+e'e\,$ is automatically invertible, so we study also the broader class of rings having such an invertible anti-commutator. Simple artinian rings $\,R\,$ (along with other related classes of matrix rings) with one of the above properties are completely determined. In this study, we also arrive at various new criteria for {\it general\} $\,2\times 2\,$ matrix rings. For instance, $R\,$ is such a matrix ring if and only if it has an invertible commutator $\,er-re\,$ where $\,e^2=e$.