Stable range one for rings with central units

摘要

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