| Title:
|
Strongly 2-nil-clean rings with involutions (English) |
| Author:
|
Chen, Huanyin |
| Author:
|
Sheibani Abdolyousefi, Marjan |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
69 |
| Issue:
|
2 |
| Year:
|
2019 |
| Pages:
|
317-330 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
A $*$-ring $R$ is strongly 2-nil-$*$-clean if every element in $R$ is the sum of two projections and a nilpotent that commute. Fundamental properties of such $*$-rings are obtained. We prove that a $*$-ring $R$ is strongly 2-nil-$*$-clean if and only if for all $a\in R$, $a^2\in R$ is strongly nil-$*$-clean, if and only if for any $a\in R$ there exists a $*$-tripotent $e\in R$ such that $a-e\in R$ is nilpotent and $ea=ae$, if and only if $R$ is a strongly $*$-clean SN ring, if and only if $R$ is abelian, $J(R)$ is nil and $R/J(R)$ is $*$-tripotent. Furthermore, we explore the structure of such rings and prove that a $*$-ring $R$ is strongly 2-nil-$*$-clean if and only if $R$ is abelian and $R\cong R_1, R_2$ or $R_1\times R_2$, where $R_1/J(R_1)$ is a $*$-Boolean ring and $J(R_1)$ is nil, $R_2/J(R_2)$ is a $*$-Yaqub ring and $J(R_2)$ is nil. The uniqueness of projections of such rings are thereby investigated. (English) |
| Keyword:
|
nilpotent |
| Keyword:
|
projection |
| Keyword:
|
$*$-tripotent ring |
| Keyword:
|
symmetry |
| Keyword:
|
strongly $*$-clean ring |
| MSC:
|
16E50 |
| MSC:
|
16U99 |
| MSC:
|
16W10 |
| idZBL:
|
Zbl 07088786 |
| idMR:
|
MR3959946 |
| DOI:
|
10.21136/CMJ.2018.0291-17 |
| . |
| Date available:
|
2019-05-24T08:54:34Z |
| Last updated:
|
2021-07-05 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/147726 |
| . |
| Reference:
|
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| . |
