Czechoslovak Mathematical Journal, Vol. 69, No. 1, pp. 197-205, 2019

Nil-clean and unit-regular elements in certain subrings of ${\mathbb M}_2(\mathbb Z)$

Yansheng Wu, Gaohua Tang, Guixin Deng, Yiqiang Zhou

Received May 25, 2017.   Published online July 23, 2018.

Abstract:  An element in a ring is clean (or, unit-regular) if it is the sum (or, the product) of an idempotent and a unit, and is nil-clean if it is the sum of an idempotent and a nilpotent. Firstly, we show that Jacobson's lemma does not hold for nil-clean elements in a ring, answering a question posed by Koşan, Wang and Zhou (2016). Secondly, we present new counter-examples to Diesl's question whether a nil-clean element is clean in a ring. Lastly, we give new examples of unit-regular elements that are not clean in a ring. The rings under consideration in our examples are particular subrings of $\mathbb{M}_2(\mathbb{Z})$.
Keywords:  clean element; nil-clean element; unit-regular element; Jacobson's lemma for nil-clean elements
Classification MSC:  16U60, 16S50, 11D09
DOI:  10.21136/CMJ.2018.0256-17

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Affiliations:   Yansheng Wu, Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing, Jiangsu, 211100, P. R. China, e-mail:; Gaohua Tang, Key Laboratory of Environment Change and Resources Use in Beibu Gulf (Guangxi Teachers Education University), Ministry of Education, P. R. China; School of Mathematics and Statistics, Guangxi Teachers Education University, Nanning, Guangxi, 530023, P. R. China; School of Sciences, Qinzhou University, Qinzhou, Guangxi 535011, P. R. China; Guixin Deng, School of Mathematics and Statistics, Guangxi Teachers Education University, Nanning, Guangxi, 530023, P. R. China; Yiqiang Zhou, Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John's, NL A1C 5S7, Canada

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