Uniform diagonalisation of matrices over regular rings. (Q1771848)

A regular ring \(R\) is separative if \(A\oplus A\simeq A\oplus B\simeq B\oplus B\) imply \(A\simeq B\) for all finitely generated projective \(R\)-modules \(A,B\). \(2\times 2\) matrices over \(R\) are uniformly diagonalised if there exist \(2\times 2\) matrices \(P,Q\) whose elements depend only on a given \(2\times 2\) matrix \(A\) such that \(PAQ\) is a diagonal matrix. It is shown that the separativity problem for von Neumann regular rings is equivalent to the existence of formula for diagonalised matrices over \(R\).