Elementary matrix reduction over Bézout domains

Abstract

A ring R is an elementary divisor ring if every matrix over R admits a diagonal reduction. If R is an elementary divisor domain, we prove that R is a Bézout duo-domain if and only if for any a ∈ R, RaR = R =⇒ ∃ s, t ∈ R such that sat = 1. We further explore various stable-like conditions on a Bézout duo-domain under which it is an elementary divisor domain. Many known results are thereby generalized to much wider class of rings, e.g. [3, Theorem 3.4.], [5, Theorem 14], [9, Theorem 3.7], [13, Theorem 4.7.1] and [14, Theorem 3].