Stable range one for rings with central units

TL;DR

This paper investigates conditions under which rings with stable range one and central units are commutative, providing partial positive answers for specific classes of rings.

Contribution

It identifies several conditions that ensure rings with stable range one and central units are commutative, advancing understanding of this algebraic property.

Findings

Rings with stable range one and central units are commutative if they are semiprime.

Such rings are also commutative if they are one-sided Noetherian.

Additional conditions include having unit-stable range 1, classical Krull dimension 0, or specific algebraic properties over fields.

Abstract

The purpose of this paper is to give a partial positive answer to a question raised by Khurana et al. as to whether a ring $R$ with stable range one and central units is commutative. We show that this is the case under any of the following additional conditions: $R$ is semiprime or $R$ is one-sided Noetherian or $R$ has unit-stable range $1$ or $R$ has classical Krull dimension $0$ or $R$ is an algebra over a field $K$ such that $K$ is uncountable and $R$ has only countably many primitive ideals or $R$ is affine and either $K$ has characteristic $0$ or has infinite transcendental degree over its prime subfield or is algebraically closed. However, the general question remains open.