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ON A QUESTION OF HARTWIG AND LUH
Published online by Cambridge University Press: 13 June 2013
Abstract
In 1977 Hartwig and Luh asked whether an element $a$ in a Dedekind-finite ring $R$ satisfying $aR= {a}^{2} R$ also satisfies $Ra= R{a}^{2} $. In this paper, we answer this question in the negative. We also prove that if $a$ is an element of a Dedekind-finite exchange ring $R$ and $aR= {a}^{2} R$, then $Ra= R{a}^{2} $. This gives an easier proof of Dischinger’s theorem that left strongly $\pi $-regular rings are right strongly $\pi $-regular, when it is already known that $R$ is an exchange ring.
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- Research Article
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- Copyright ©2013 Australian Mathematical Publishing Association Inc.
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