| Title:
|
Maps on upper triangular matrices preserving zero products (English) |
| Author:
|
Słowik, Roksana |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
67 |
| Issue:
|
4 |
| Year:
|
2017 |
| Pages:
|
1095-1103 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Consider $\mathcal T_n(F)$---the ring of all $n\times n$ upper triangular matrices defined over some field $F$. A map $\phi $ is called a zero product preserver on ${\mathcal T}_n(F)$ in both directions if for all $x,y\in {\mathcal T}_n(F)$ the condition $xy=0$ is satisfied if and only if $\phi (x)\phi (y)=0$. In the present paper such maps are investigated. The full description of bijective zero product preservers is given. Namely, on the set of the matrices that are invertible, the map $\phi $ may act in any bijective way, whereas for the zero divisors and zero matrix one can write $\phi $ as a composition of three types of maps. The first of them is a conjugacy, the second one is an automorphism induced by some field automorphism, and the third one transforms every matrix $x$ into a matrix $x'$ such that $\{y\in \mathcal T_n(F)\colon xy=0\}=\{y\in \mathcal T_n(F)\colon x'y=0\}$, $\{y\in \mathcal T_n(F)\colon yx=0\}=\{y\in \mathcal T_n(F)\colon yx'=0\}$. (English) |
| Keyword:
|
zero product preserver |
| Keyword:
|
upper triangular matrix |
| MSC:
|
15A99 |
| MSC:
|
16U99 |
| idZBL:
|
Zbl 06819575 |
| idMR:
|
MR3736021 |
| DOI:
|
10.21136/CMJ.2017.0416-16 |
| . |
| Date available:
|
2017-11-20T14:57:44Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/146969 |
| . |
| Reference:
|
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