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

Following O'Meara's result Journal of Algebra and Its Applications Vol~13, No. 8 (2014), it follows that the block matrix A=pmatrix B & 0 0 & 0 pmatrix M₍+ₑ (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: 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=pmatrix B & 0 0 & 0 pmatrix M₍+ₑ (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.