Factorizations of Matrices Over Projective-free Rings

TL;DR

This paper characterizes when matrices over projective-free rings are strongly J#-clean, extending known results on strongly clean matrices over commutative local rings.

Contribution

It provides a characterization of strongly J#-clean matrices over projective-free rings, broadening the understanding of matrix decompositions in non-commutative ring settings.

Findings

Characterization of strongly J#-clean matrices over projective-free rings

Extension of results from commutative local rings to projective-free rings

Broader understanding of matrix decompositions in ring theory

Abstract

An element of a ring $R$ is called strongly $J^{\#}$-clean provided that it can be written as the sum of an idempotent and an element in $J^{\#}(R)$ that commute. We characterize, in this article, the strongly $J^{\#}$-cleanness of matrices over projective-free rings. These extend many known results on strongly clean matrices over commutative local rings.