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ON A QUESTION OF HARTWIG AND LUH

Part of: Conditions on elements Homological methods

Published online by Cambridge University Press:  13 June 2013

SAMUEL J. DITTMER ,
DINESH KHURANA  and
PACE P. NIELSEN
SAMUEL J. DITTMER
Affiliation:
Department of Mathematics, Brigham Young University, Provo, UT 84602, USA email samuel.dittmer@gmail.com
DINESH KHURANA
Affiliation:
Department of Mathematics, Panjab University, Chandigarh 160 014, India email dkhurana@pu.ac.in
PACE P. NIELSEN*
Affiliation:
Department of Mathematics, Brigham Young University, Provo, UT 84602, USA
*
✉pace@math.byu.edu
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Abstract

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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.

Keywords

Dedekind-finite ringsDischinger’s theoremexchange ringsstrongly π-regular rings

MSC classification

Secondary: 16E50: von Neumann regular rings and generalizations 16U99: None of the above, but in this section
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 89 , Issue 2 , April 2014 , pp. 271 - 278
Copyright
Copyright ©2013 Australian Mathematical Publishing Association Inc. 

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