| Title:
|
$(\phi , \varphi )$-derivations on semiprime rings and Banach algebras (English) |
| Author:
|
Wani, Bilal Ahmad |
| Language:
|
English |
| Journal:
|
Communications in Mathematics |
| ISSN:
|
1804-1388 (print) |
| ISSN:
|
2336-1298 (online) |
| Volume:
|
29 |
| Issue:
|
3 |
| Year:
|
2021 |
| Pages:
|
371-383 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Let $\mathcal{R} $ be a semiprime ring with unity $e$ and $\phi $, $\varphi $ be automorphisms of $\mathcal{R} $. In this paper it is shown that if $\mathcal{R} $ satisfies $$2\mathcal{D} (x^n) = \mathcal{D} (x^{n-1})\phi (x) + \varphi (x^{n-1})\mathcal{D} (x)+\mathcal{D} (x)\phi (x^{n-1}) + \varphi (x)\mathcal{D} (x^{n-1})$$ for all $x\in \mathcal{R} $ and some fixed integer $n\geq 2$, then $\mathcal{D} $ is an ($\phi $, $\varphi $)-derivation. Moreover, this result makes it possible to prove that if $\mathcal { R}$ admits an additive mappings $\mathcal{D} ,\mathcal{G} \colon \mathcal{R} \rightarrow \mathcal{R} $ satisfying the relations \begin {gather*}\nonumber 2\mathcal{D} (x^n) = \mathcal{D} (x^{n-1})\phi (x) + \varphi (x^{n-1})\mathcal{G} (x)+\mathcal{G} (x)\phi (x^{n-1}) + \varphi (x)\mathcal{G} (x^{n-1})\,, \\ 2\mathcal{G} (x^n) = \mathcal{G} (x^{n-1})\phi (x) + \varphi (x^{n-1})\mathcal{D} (x)+\mathcal{D} (x)\phi (x^{n-1}) + \varphi (x)\mathcal{D} (x^{n-1})\,, \end {gather*} for all $x\in \mathcal{R} $ and some fixed integer $n\geq 2$, then $\mathcal{D} $ and $\mathcal{G} $ are ($\phi $, $\varphi $)\HH derivations under some torsion restriction. Finally, we apply these purely ring theoretic results to semi-simple Banach algebras. (English) |
| Keyword:
|
Prime ring |
| Keyword:
|
semiprime ring |
| Keyword:
|
Banach algebra |
| Keyword:
|
Jordan derivation |
| Keyword:
|
$(\phi, \varphi )$-derivation |
| MSC:
|
16N60 |
| MSC:
|
16W25 |
| MSC:
|
46J10 |
| idZBL:
|
Zbl 07484374 |
| idMR:
|
MR4355419 |
| . |
| Date available:
|
2022-01-10T10:01:47Z |
| Last updated:
|
2022-04-28 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/149323 |
| . |
| Reference:
|
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