| Title:
|
Rings consisting entirely of certain elements (English) |
| Author:
|
Chen, Huanyin |
| Author:
|
Sheibani, Marjan |
| Author:
|
Ashrafi, Nahid |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
68 |
| Issue:
|
2 |
| Year:
|
2018 |
| Pages:
|
553-558 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We completely determine when a ring consists entirely of weak idempotents, units and nilpotents. We prove that such ring is exactly isomorphic to one of the following: a Boolean ring; $\Bbb Z_3\oplus {\Bbb Z}_3$; $\Bbb Z_3\oplus B$ where $B$ is a Boolean ring; local ring with nil Jacobson radical; $M_2(\Bbb Z_2)$ or $M_2(\Bbb Z_3)$; or the ring of a Morita context with zero pairings where the underlying rings are $\Bbb Z_2$ or $\Bbb Z_3$. (English) |
| Keyword:
|
idempotent |
| Keyword:
|
nilpotent |
| Keyword:
|
Boolean ring |
| Keyword:
|
local ring |
| Keyword:
|
Morita context |
| MSC:
|
16E50 |
| MSC:
|
16S34 |
| MSC:
|
16U10 |
| idZBL:
|
Zbl 06890389 |
| idMR:
|
MR3819190 |
| DOI:
|
10.21136/CMJ.2018.0554-16 |
| . |
| Date available:
|
2018-06-11T10:57:45Z |
| Last updated:
|
2020-07-06 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/147235 |
| . |
| Reference:
|
[1] Ahn, M. S., Anderson, D. D.: Weakly clean rings and almost clean rings.Rocky Mountain J. Math. 36 (2006), 783-798. Zbl 1131.13301, MR 2254362, 10.1216/rmjm/1181069429 |
| Reference:
|
[2] Anderson, D. D., Camillo, V. P.: Commutative rings whose elements are a sum of a unit and idempotent.Comm. Algebra 30 (2002), 3327-3336. Zbl 1083.13501, MR 1914999, 10.1081/agb-120004490 |
| Reference:
|
[3] Breaz, S., Gălugăreanu, 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 |
| Reference:
|
[5] Danchev, P. V., McGovern, W. W.: Commutative weakly nil clean unital rings.J. Algebra 425 (2015), 410-422. Zbl 1316.16028, MR 3295991, 10.1016/j.jalgebra.2014.12.003 |
| 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] Du, X.: The adjoint semigroup of a ring.Commun. Algebra 30 (2002), 4507-4525. Zbl 1030.16012, MR 1936488, 10.1081/AGB-120013336 |
| Reference:
|
[8] Immormino, N. A.: Clean Rings & Clean Group Rings, Ph.D. Thesis.Bowling Green State University, Bowling Green (2013). MR 3321928 |
| Reference:
|
[9] Kosan, 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:
|
[10] McGovern, W., Raja, S., Sharp, A.: Commutative nil clean group rings.J. Algebra Appl. 14 (2015). Zbl 1325.16024, MR 3338090, 10.1142/S0219498815500942 |
| Reference:
|
[11] Nicholson, W. K.: Rings whose elements are quasi-regular or regular.Aequations Math. 9 (1973), 64-70. Zbl 0255.16006, MR 0316497, 10.1007/BF01838190 |
| . |
