| Title:
|
Certain additive decompositions in a noncommutative ring (English) |
| Author:
|
Chen, Huanyin |
| Author:
|
Sheibani, Marjan |
| Author:
|
Bahmani, Rahman |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
72 |
| Issue:
|
4 |
| Year:
|
2022 |
| Pages:
|
1217-1226 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We determine when an element in a noncommutative ring is the sum of an idempotent and a radical element that commute. We prove that a $2\times 2$ matrix $A$ over a projective-free ring $R$ is strongly $J$-clean if and only if $A\in J (M_2(R))$, or $I_2-A\in J(M_2(R))$, or $A$ is similar to $\left (\smallmatrix 0&\lambda \\ 1&\mu \endsmallmatrix \right )$, where $\lambda \in J(R)$, $\mu \in 1+J(R)$, and the equation $x^2-x\mu -\lambda =0$ has a root in $J(R)$ and a root in $1+J(R)$. We further prove that $f(x)\in R[[x]]$ is strongly $J$-clean if $f(0)\in R$ be optimally $J$-clean. (English) |
| Keyword:
|
idempotent matrix |
| Keyword:
|
nilpotent matrix |
| Keyword:
|
projective-free ring |
| Keyword:
|
quadratic equation |
| Keyword:
|
power series |
| MSC:
|
15A09 |
| MSC:
|
16E50 |
| MSC:
|
16U60 |
| idZBL:
|
Zbl 07655796 |
| idMR:
|
MR4517609 |
| DOI:
|
10.21136/CMJ.2022.0039-22 |
| . |
| Date available:
|
2022-11-28T11:44:54Z |
| Last updated:
|
2025-01-06 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/151143 |
| . |
| Reference:
|
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| Reference:
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| Reference:
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| Reference:
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| Reference:
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| Reference:
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| Reference:
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| Reference:
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| Reference:
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| . |
