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.

Key Words:

• von Neumann regularity
• Separativity
• Uniform diagonalisation
• One-accessible
• Algebraicity

Mathematics Subject Classification:

• 16E50

Acknowledgments

The first author gratefully acknowledges support from the University of Canterbury through an Erskine Fellowship. Some of this work was done during a visit by the second author to Concordia University and he thanks the University for its support and hospitality. The third author is supported by the NSERC of Canada. We are grateful for suggestions from P. Ara, D. Herbera, and K. Goodearl.

Notes

^#Communicated by A. Facchini.

Deceased in March 2004. At the suggestion of the first author we dedicate this paper to the late D.V. Tyukavkin.