2024/07/17 by S. K. Jain, André Leroy, Jain, S. K. +1
Engineering · Mathematics · #15A23 #15B33 #16E30 #Advanced Topics in Algebra #FOS: Mathematics #Rings and Algebras (math.RA) #Rings, Modules, and Algebras #graph theory and CDMA systems
paper · pdf · doi:10.48550/arxiv.2407.12231
openalex publication_date 2024/07/17 · openalex created_date 2024/09/09 · openalex updated_date 2026/07/28
Following O'Meara's result [Journal of Algebra and Its Applications Vol~13, No. 8 (2014)], it follows that the block matrix A=\beginpmatrix B & 0 0 & 0 \endpmatrix ∈ Mn+r(R), B∈ Mn(R), r≥ 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=\beginpmatrix B & 0 0 & 0 \endpmatrix ∈ Mn+r(R) where B∈ Mn(R), r≥ 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.