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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Connections between unit-regularity, regularity, cleanness, and strong cleanness of elements and rings

Authors: Pace P. Nielsen and Janez Šter
Journal: Trans. Amer. Math. Soc. 370 (2018), 1759-1782
MSC (2010): Primary 16E50; Secondary 16D70, 16S50, 16U99
Published electronically: November 16, 2017
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Abstract: We construct an example of a unit-regular ring which is not
strongly clean, answering an open question of Nicholson. We also characterize clean matrices with a zero column, and this allows us to describe an interesting connection between unit-regular elements and clean elements. Next we study in arbitrary rings those elements whose powers are regular, and provide a method for constructing inner inverses which satisfy many additional strong relations. As a corollary we show that if each of the powers $ a,a^2,\ldots , a^n$ is a regular element in some ring $ R$ (for some $ n\geq 1$), then there exists $ w\in R$ such that $ a^k w^k a^k=a^k$ and $ w^k a^k w^k=w^k$ for $ 1\leq k\leq n$. Similar statements are also obtained for unit-regular elements. The paper ends with a large number of examples elucidating further connections (and disconnections) between cleanness, regularity, and unit-regularity.

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Additional Information

Pace P. Nielsen
Affiliation: Department of Mathematics, Brigham Young University, Provo, Utah 84602

Janez Šter
Affiliation: Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 21, 1000 Ljubljana, Slovenia

Keywords: (Strongly) clean element/ring, (unit-)regular element/ring
Received by editor(s): October 19, 2015
Received by editor(s) in revised form: May 30, 2016
Published electronically: November 16, 2017
Additional Notes: This work was partially supported by a grant from the Simons Foundation (#315828 to the first author). The project was sponsored by the National Security Agency under Grant No. H98230-16-1-0048.
Article copyright: © Copyright 2017 American Mathematical Society

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