Matrices over a commutative ring as sums of three idempotents or three involutions

TL;DR

This paper investigates the conditions under which matrices over a commutative ring can be expressed as sums of three idempotent or involutive matrices, extending previous results from fields and specific rings.

Contribution

It generalizes the decomposition of matrices into sums of three idempotents or involutions from fields to arbitrary commutative rings.

Findings

Matrices over a commutative ring can be decomposed into sums of three idempotents.

Matrices over a commutative ring can be decomposed into sums of three involutions.

The work extends known results from fields to more general rings.

Abstract

Motivated by Hirano-Tominaga's work \cite{HT} on rings for which every element is a sum of two idempotents and by de Seguins Pazzis's results \cite{de} on decomposing every matrix over a field of positive characteristic as a sum of idempotent matrices, we address decomposing every matrix over a commutative ring as a sum of three idempotent matrices and, respectively, as a sum of three involutive matrices.