| Title:
|
Nil-clean and unit-regular elements in certain subrings of ${\mathbb M}_2(\mathbb Z)$ (English) |
| Author:
|
Wu, Yansheng |
| Author:
|
Tang, Gaohua |
| Author:
|
Deng, Guixin |
| Author:
|
Zhou, Yiqiang |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
69 |
| Issue:
|
1 |
| Year:
|
2019 |
| Pages:
|
197-205 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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 $\mathbb {M}_2(\mathbb {Z})$. (English) |
| Keyword:
|
clean element |
| Keyword:
|
nil-clean element |
| Keyword:
|
unit-regular element |
| Keyword:
|
Jacobson's lemma for nil-clean elements |
| MSC:
|
11D09 |
| MSC:
|
16S50 |
| MSC:
|
16U60 |
| idZBL:
|
Zbl 07088779 |
| idMR:
|
MR3923584 |
| DOI:
|
10.21136/CMJ.2018.0256-17 |
| . |
| Date available:
|
2019-03-08T15:00:02Z |
| Last updated:
|
2021-04-05 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/147627 |
| . |
| Reference:
|
[1] Andrica, D., Călugăreanu, G.: A nil-clean $2 \times 2$ matrix over the integers which is not clean.J. Algebra Appl. 13 (2014), Article ID 1450009, 9 pages. Zbl 1294.16019, MR 3195166, 10.1142/S0219498814500091 |
| Reference:
|
[2] Camillo, V. P., Khurana, D.: A characterization of unit regular rings.Commun. Algebra 29 (2001), 2293-2295. Zbl 0992.16011, MR 1837978, 10.1081/AGB-100002185 |
| Reference:
|
[3] Chen, J., Yang, X., Zhou, Y.: On strongly clean matrix and triangular matrix rings.Commun. Algebra 34 (2006), 3659-3674. Zbl 1114.16024, MR 2262376, 10.1080/00927870600860791 |
| Reference:
|
[4] Cvetkovic-Ilic, D., Harte, R.: On Jacobson's lemma and Drazin invertibility.Appl. Math. Lett. 23 (2010), 417-420. Zbl 1195.16033, MR 2594854, 10.1016/j.aml.2009.11.009 |
| Reference:
|
[5] Diesl, A. J.: Nil clean rings.J. Algebra 383 (2013), 197-211. Zbl 1296.16016, MR 3037975, 10.1016/j.jalgebra.2013.02.020 |
| Reference:
|
[6] Khurana, D., Lam, T. Y.: Clean matrices and unit-regular matrices.J. Algebra 280 (2004), 683-698. Zbl 1067.16050, MR 2090058, 10.1016/j.jalgebra.2004.04.019 |
| Reference:
|
[7] Koşan, T., Wang, Z., Zhou, Y.: Nil-clean and strongly nil-clean rings.J. Pure Appl. Algebra 220 (2016), 633-646. Zbl 1335.16026, MR 3399382, 10.1016/j.jpaa.2015.07.009 |
| Reference:
|
[8] Lam, T. Y., Nielsen, P. P.: Jacobson's lemma for Drazin inverses.Ring Theory and Its Applications D. V. Huynh et al. Contemporary Mathematics 609, American Mathematical Society, Providence (2014), 185-195. Zbl 1294.15005, MR 3204360, 10.1090/conm/609/12127 |
| Reference:
|
[9] Tang, G., Zhou, Y.: A class of formal matrix rings.Linear Algebra Appl. 438 (2013), 4672-4688. Zbl 1283.16026, MR 3039217, 10.1016/j.laa.2013.02.019 |
| Reference:
|
[10] Zhuang, G., Chen, J., Cui, J.: Jacobson's lemma for the generalized Drazin inverse.Linear Algebra Appl. 436 (2012), 742-746. Zbl 1231.15008, MR 2854904, 10.1016/j.laa.2011.07.044 |
| . |
