Diagonal reduction of matrices over commutative semihereditary Bezout rings

TL;DR

This paper proves that certain commutative semihereditary Bezout rings with Gelfand elements are elementary divisor rings, advancing the understanding of matrix reduction in algebraic structures.

Contribution

It establishes that such rings are elementary divisor rings, providing a new criterion for matrix diagonalization over these rings.

Findings

Every commutative semihereditary Bezout ring with Gelfand elements is an elementary divisor ring.

The result extends the class of rings known to have elementary divisor properties.

Provides a new algebraic condition linking Gelfand elements to matrix reduction capabilities.

Abstract

It is proven that every commutative semihereditary Bezout ring in which any regular element is Gelfand (adequate), is an elementary divisor ring.