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
Affiliations: Yansheng Wu, Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing, Jiangsu, 211100, P. R. China, e-mail: wysasd@163.com; 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
