On Uniform Diagonalisation of Matrices over Regular Rings and One-Accessible Regular Algebras#

Abstract

In connection with the fundamental Separativity Problem for regular rings, we show that a regular algebra R over a commutative ring admits a uniform diagonalisation formula where the entries of P and Q are algebra expressions in the a i and the a i ', if and only if R is strongly regular (abelian regular in the terminology of Goodearl, K.R. (1979). Von Neumann Regular Rings. London: Pitman. 2nd ed. Krieger, Malabar, CFI. 1991). Next, we study regular algebras R over a field F such that for any a ∈ R there exist b ∈ F[a] and b' ∈ R such that bb'b = b, b'bb' = b' and the subalgebra of R generated by a and b' is regular. Such algebras are called one-accessible. We show that a finite product of matrix rings over a field is one-accessible and that a regular algebra over an uncountable perfect field is one-accessible if and only if it is algebraic. Tangentially, we elucidate and characterize when a nilpotent element has all its powers regular (or unit-regular) in an arbitrary algebra R over a commutative ring Λ. This involves finite direct products of matrix rings over factor rings of Λ.