Refinement rings, localization and diagonal reduction of matrices

TL;DR

This paper explores the properties of refinement rings, showing that module isomorphism can be checked locally at maximal ideals, and establishes conditions for diagonal reduction of regular matrices over such rings.

Contribution

It proves that module isomorphism over a refinement ring can be characterized by localizations at maximal ideals and links diagonal reduction of matrices over the ring and its quotient.

Findings

Module isomorphism is determined by localizations at maximal ideals.

Diagonal reduction of matrices over the ring is equivalent to that over its quotient.

Regular matrices over the quotient admit diagonal reduction iff over the ring.

Abstract

In this paper we prove that if $R$ is a commutative refinement ring and $M$, $N$ are two $R$-modules then, $M\cong N$ if and only if for every maximal ideal $m$ of $R$, $M_m\cong N_m$. We prove if $R$ is a refinement ring, then every regular matrix over $\frac{R}{J(R)}$ admits a diagonal reduction iff every regular matrix over $R$ admits a diagonal reduction.