Elementary Matrix Reduction Over J-Stable Rings

TL;DR

This paper characterizes J-stable rings as elementary divisor rings precisely when they are Bezout rings, linking ring stability properties with matrix reduction capabilities.

Contribution

It establishes a necessary and sufficient condition for J-stable rings to be elementary divisor rings, connecting stability, Bezout property, and matrix diagonalization.

Findings

J-stable rings are elementary divisor rings iff they are Bezout rings.

Provides a characterization linking stability and matrix reduction.

Enhances understanding of ring structures for matrix diagonalization.

Abstract

A commutative ring $R$ is J-stable provided that for any $a\not\in J(R)$, $R/aR$ has stable range one. A ring $R$ is called an elementary divisor ring if every $m\times n$ matrix over $R$ admits diagonal reduction. We prove that a J-stabe ring $R$ is an elementary divisor ring if and only if it is a Bezout ring.