| Title:
|
On feebly nil-clean rings (English) |
| Author:
|
Sheibani Abdolyousefi, Marjan |
| Author:
|
Pouyan, Neda |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
74 |
| Issue:
|
1 |
| Year:
|
2024 |
| Pages:
|
87-94 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
A ring $R$ is feebly nil-clean if for any $a\in R$ there exist two orthogonal idempotents $e,f\in R$ and a nilpotent $w\in R$ such that $a=e-f+w$. Let $R$ be a 2-primal feebly nil-clean ring. We prove that every matrix ring over $R$ is feebly nil-clean. The result for rings of bounded index is also obtained. These provide many classes of rings over which every matrix is the sum of orthogonal idempotent and nilpotent matrices. (English) |
| Keyword:
|
orthogonal idempotent matrix |
| Keyword:
|
nilpotent matrix |
| Keyword:
|
matrix ring |
| Keyword:
|
feebly nil-clean ring |
| MSC:
|
15A23 |
| MSC:
|
15B33 |
| MSC:
|
16U99 |
| idZBL:
|
Zbl 07893368 |
| idMR:
|
MR4717823 |
| DOI:
|
10.21136/CMJ.2023.0215-22 |
| . |
| Date available:
|
2024-03-13T10:04:05Z |
| Last updated:
|
2026-04-06 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/152269 |
| . |
| Reference:
|
[1] Abyzov, A. N., Mukhametgaliev, I. I.: On some matrix analogs of the little Fermat theorem.Math. Notes 101 (2017), 187-192. Zbl 1365.16024, MR 3608014, 10.1134/S0001434617010229 |
| Reference:
|
[2] Arora, N., Kundu, S.: Commutative feebly clean rings.J. Algebra Appl. 16 (2017), Article ID 1750128, 14 pages. Zbl 1368.13006, MR 3660411, 10.1142/S0219498817501286 |
| Reference:
|
[3] Breaz, S., Călugăreanu, G., Danchev, P., Micu, T.: Nil-clean matrix rings.Linear Algebra Appl. 439 (2013), 3115-3119. Zbl 1355.16023, MR 3116417, 10.1016/j.laa.2013.08.027 |
| Reference:
|
[4] Chen, H.: Rings Related Stable Range Conditions.Series in Algebra 11. World Scientific, Hackensack (2011). Zbl 1245.16002, MR 2752904, 10.1142/8006 |
| Reference:
|
[5] Chen, H., Sheibani, M.: Strongly 2-nil-clean rings.J. Algebra Appl. 16 (2017), Article ID 1750178, 12 pages. Zbl 1382.16035, MR 3661645, 10.1142/S021949881750178X |
| Reference:
|
[6] 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:
|
[7] Han, J., Nicholson, W. K.: Extensions of clean rings.Commun. Algebra 29 (2001), 2589-2595. Zbl 0989.16015, MR 1845131, 10.1081/AGB-100002409 |
| Reference:
|
[8] Hirano, Y., Tominaga, H.: Rings in which every element is a sum of two idempotents.Bull. Aust. Math. Soc. 37 (1988), 161-164. Zbl 0688.16015, MR 0930784, 10.1017/S000497270002668X |
| Reference:
|
[9] Hirano, Y., Tominaga, H., Yaqub, A.: On rings in which every element is uniquely expressible as a sum of a nilpotent element and a certain potent element.Math. J. Okayama Univ. 30 (1988), 33-40. Zbl 0665.16016, MR 0976729, 10.18926/mjou/33546 |
| Reference:
|
[10] Koşan, M. T., Lee, T.-K., Zhou, Y.: When is every matrix over a division ring a sum of an idempotent and a nilpotent?.Linear Algebra Appl. 450 (2014), 7-12. Zbl 1303.15016, MR 3192466, 10.1016/j.laa.2014.02.047 |
| Reference:
|
[11] 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:
|
[12] Tominaga, H., Yaqub, A.: On generalized $n$-like rings and related rings.Math. J. Okayama Univ. 23 (1981), 199-202. Zbl 0477.16018, MR 0638143 |
| Reference:
|
[13] Ying, Z., Koşan, T., Zhou, Y.: Rings in which every element is a sum of two tripotents.Can. Math. Bull. 59 (2016), 661-672. Zbl 1373.16067, MR 3563747, 10.4153/CMB-2016-009-0 |
| Reference:
|
[14] Yu, H.-P.: On quasi-duo rings.Glasg. Math. J. 37 (1995), 21-31. Zbl 0819.16001, MR 1316960, 10.1017/S0017089500030342 |
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