Full Text: PDF
DOI: 10.23952/cot.2025.36
Received May 22, 2024; Accepted July 17, 2024; Published online June 19, 2025
Abstract.Following O’Meara’s result [Journal of Algebra and Its Applications, 13 (2014) 8], it follows that the block matrix
$latex A=\begin{pmatrix}
B & 0 \\
0 & 0
\end{pmatrix} \in M_{n+r}(R)$, 
, 
, over a von Neumann regular separative ring 
, is a product of idempotent matrices. Furthermore, this decomposition into idempotents of 
also holds when $atex B$ is an invertible matrix and is a GE ring (defined by Cohn [New mathematical monographs: 3, Cambridge University Press, 2006]). As a consequence, it follows that if there exists an example of a von Neumann regular ring over which the matrix $latex A=\begin{pmatrix}
B & 0 \\
0 & 0
\end{pmatrix}\in M_{n+r}(R),$ where , , cannot be expressed as a product of idempotents, then 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 a counter example related to an open question whether every totally nonnegative matrix is a product of nonnegative idempotent matrices.
How to Cite this Article:
S.K. Jain, A. Leroy, Decomposition of matrices into product of idempotents and separativity of regular rings, Commun. Optim. Theory 2025 (2025) 36.