Czechoslovak Mathematical Journal, first online, pp. 1-9


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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References:
[1] D. Andrica, G. Călugăreanu: A nil-clean $2 \times 2$ matrix over the integers which is not clean. J. Algebra Appl. 13 (2014), Article ID 1450009, 9 pages. DOI 10.1142/S0219498814500091 | MR 3195166 | Zbl 1294.16019
[2] V. P. Camillo, D. Khurana: A characterization of unit regular rings. Commun. Algebra 29 (2001), 2293-2295. DOI 10.1081/AGB-100002185 | MR 1837978 | Zbl 0992.16011
[3] J. Chen, X. Yang, Y. Zhou: On strongly clean matrix and triangular matrix rings. Commun. Algebra 34 (2006), 3659-3674. DOI 10.1080/00927870600860791 | MR 2262376 | Zbl 1114.16024
[4] D. Cvetkovic-Ilic, R. Harte: On Jacobson's lemma and Drazin invertibility. Appl. Math. Lett. 23 (2010), 417-420. DOI 10.1016/j.aml.2009.11.009 | MR 2594854 | Zbl 1195.16033
[5] A. J. Diesl: Nil clean rings. J. Algebra 383 (2013), 197-211. DOI 10.1016/j.jalgebra.2013.02.020 | MR 3037975 | Zbl 1296.16016
[6] D. Khurana, T. Y. Lam: Clean matrices and unit-regular matrices. J. Algebra 280 (2004), 683-698. DOI 10.1016/j.jalgebra.2004.04.019 | MR 2090058 | Zbl 1067.16050
[7] T. Koşan, Z. Wang, Y. Zhou: Nil-clean and strongly nil-clean rings. J. Pure Appl. Algebra 220 (2016), 633-646. DOI 10.1016/j.jpaa.2015.07.009 | MR 3399382 | Zbl 1335.16026
[8] T. Y. Lam, P. P. Nielsen: Jacobson's lemma for Drazin inverses. Ring Theory and Its Applications (D. V. Huynh et al., eds.). Contemporary Mathematics 609, American Mathematical Society, Providence, 2014, pp. 185-195. DOI 10.1090/conm/609/12127 | MR 3204360 | Zbl 1294.15005
[9] G. Tang, Y. Zhou: A class of formal matrix rings. Linear Algebra Appl. 438 (2013), 4672-4688. DOI 10.1016/j.laa.2013.02.019 | MR 3039217 | Zbl 1283.16026
[10] G. Zhuang, J. Chen, J. Cui: Jacobson's lemma for the generalized Drazin inverse. Linear Algebra Appl. 436 (2012), 742-746. DOI 10.1016/j.laa.2011.07.044 | MR 2854904 | Zbl 1231.15008

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


 
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