Stable range one for rings with central units

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.

Keywords:

• Algebras over fields
• associative rings
• central units
• stable range one

MATHEMATICAL SUBJECT CLASSIFICATION:

• Primary 16U60, 16U70
• Secondary 19B10

Acknowledgment

The authors would like to thank the referee for shortening the proof of Theorem 4.1(iii). The first two named authors were partially supported by CMUP (UID/MAT/00144/2019), which is funded by FCT with national (MCTES) and European structural funds through the program FEDER, under the partnership agreement PT2020. A part of this work was done while the third named author visited the University of Porto. He would like to thank the University for hospitality and good working conditions.