Elementary Matrix Reduction Over Certain Rings

TL;DR

This paper investigates the conditions under which certain rings allow elementary matrix reduction, establishing a key equivalence between being an elementary divisor ring and a Bezout ring for locally stable rings, and extending known domain results to broader classes.

Contribution

It proves that locally stable rings are elementary divisor rings if and only if they are Bezout rings, extending classical results to rings with zero divisors.

Findings

Locally stable rings are elementary divisor rings iff they are Bezout rings.

Elementary matrix reduction properties are studied over localizations of rings.

Results extend known domain theorems to more general commutative rings.

Abstract

We explore elementary matrix reduction over certain rings characterized by their localizations. Let $R$ be a locally stable ring, we prove that $R$ is an elementary divisor ring if and only if $R$ is a Bezout ring. Elementary matrix reduction over some related localizations are also studied. Many known results on domains are thereby extended to general commutative rings which may contain many zero divisors.