Decomposition of matrices into product of idempotents and separativity of regular rings

TL;DR

This paper investigates conditions under which matrices over regular rings can be decomposed into products of idempotent matrices, linking such decompositions to the separativity property of rings and addressing an open question in ring theory.

Contribution

It extends known results on matrix decompositions over regular rings, establishing a connection between idempotent factorizations and the separativity of rings, and provides insights into an open problem about nonnegative matrices.

Findings

Matrices over separative regular rings can be decomposed into idempotents.

If certain matrices cannot be expressed as products of idempotents, the ring is not separative.

Open question remains whether all totally nonnegative matrices are products of nonnegative idempotents.

Abstract

Following O'Meara's result [Journal of Algebra and Its Applications Vol~\textbf{13}, No. 8 (2014)], it follows that the block matrix $A=\begin{pmatrix} B & 0 0 & 0 \end{pmatrix} \in M_{n+r}(R)$, $B\in M_n(R)$, $r\ge 1$, over a von Neumann regular separative ring $R$, is a product of idempotent matrices. Furthermore, this decomposition into idempotents of $A$ also holds when $B$ is an invertible matrix and $R$ is a GE ring (defined by Cohn [New mathematical monographs: {\bf 3}, Cambridge University Press (2006)]). As a consequence, it follows that if there exists an example of a von Neumann regular ring $R$ over which the matrix $ A=\begin{pmatrix} B & 0 0 & 0 \end{pmatrix} \in M_{n+r}(R) $ where $B\in M_n(R)$, $r\ge 1$ , cannot be expressed as a product of idempotents, then $R$ is not separative, thus providing an answer to an open question whether there exists a von Neumann regular ring which is not separative. The paper concludes with an example of an open question whether every totally nonnegative matrix is a product of nonnegative idempotent matrices.