Recent zbMATH articles in MSC 19Bhttps://www.zbmath.org/atom/cc/19B2021-02-12T15:23:00+00:00WerkzeugStable range one for rings with central units.https://www.zbmath.org/1452.160342021-02-12T15:23:00+00:00"Carvalho, Paula A. A. B."https://www.zbmath.org/authors/?q=ai:carvalho.paula-a-a-b"Lomp, Christian"https://www.zbmath.org/authors/?q=ai:lomp.christian"Matczuk, Jerzy"https://www.zbmath.org/authors/?q=ai:matczuk.jerzyA ring \(R\) is called by Khurana-Marks-Srivastava in [\textit{D. Khurana} et al., in: Advances in ring theory. Papers of the conference on algebra and applications, Athens, OH, USA, June 18--21, 2008. Basel: BirkhĂ¤user. 205--212 (2010; Zbl 1202.16033)] \textit{unit-central} if all its units are central. Moreover, they asked whether a unit-central ring \(R\) with stable range one is always commutative.
In the paper under review, it is shown that this is true in any of the following additional conditions:
\begin{itemize}
\item[(1)] \(R\) is semiprime, or
\item[(2)] \(R\) is one-sided Noetherian, or
\item[(3)] \(R\) has unit-stable range \(1\), or
\item[(4)] \(R\) has classical Krull dimension \(0\), or
\item[(5)] \(R\) is an algebra over a field \(K\) such that \(K\) is uncountable and \(R\) has only countably many primitive ideals, or
\item[(6)] \(R\) is affine and either \(K\) has characteristic \(0\) or has infinite transcendental degree over its prime subfield or is algebraically closed (see, respectively, Theorems 3.1, 4.1 and 4.7).
\end{itemize}
Reviewer: Peter Danchev (Sofia)