Commutative Bezout domains of stable range 1.5

TL;DR

This paper studies commutative Bezout domains with stable range 1.5, showing that matrices over such rings can be simplified to Smith's form with specific invertible transformations, and generalizes Helmer's theorem.

Contribution

It extends Smith's canonical form results to commutative Bezout domains of stable range 1.5 and generalizes Helmer's theorem on gcd of matrix entries.

Findings

Matrices over these rings can be reduced to Smith's form with elementary transformations.

All finitely generated ideals in the ring are principal.

Generalization of Helmer's theorem on gcd of matrix entries.

Abstract

A ring R is said to be of stable range 1.5 if for each a, b from R and nonzero c from R satisfying aR + bR + cR = R there exists r from R such that (a + br)R + cR = R. Let R be a commutative domain in which all finitely generated ideals are principal, and let R be of stable range 1.5. Then each matrix A over R is reduced to Smith's canonical form by transformations PAQ in which P and Q are invertible and at least one of them can be chosen to be a product of elementary matrices. We generalize Helmer's theorem about the greatest common divisor of entries of A over R.