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arXiv · 1910.04410

Stable range one for rings with central units

Paula A. A. B. Carvalho·Christian Lomp·Jerzy Matczuk
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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.

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Paula A. A. B. Carvalho, Christian Lomp, Jerzy Matczuk. 2019-10-10. Stable range one for rings with central units. https://arxiv.org/abs/1910.04410

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