Elementary matrix reduction over certain rings

Abstract

A commutative ring R is J-stable provided that R/aR has stable range 1 for all a ∈ J(R). A commutative ring R in which every finitely generated ideal is principal is called a Bézout ring. A ring R is an elementary divisor ring provided that every matrix over R admits a diagonal reduction. We prove that a J-stable ring is a Bézout ring if and only if it is an elementary divisor ring. Further, we prove that every J-stable ring is strongly completable. Various types of J-stable rings are provided. Many known results are thereby generalized to much wider class of rings, e.g.

Conditions for stable range of an elementary divisor rings

“…[6] The S(R) is saturated and multiplicatively closed.Proof. Let a, b ∈ S. By[1], stable range of R/(aR ∩ bR) is equal to 1.…”