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

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.