| Title:
|
Generalized derivations acting on multilinear polynomials in prime rings (English) |
| Author:
|
Dhara, Basudeb |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
68 |
| Issue:
|
1 |
| Year:
|
2018 |
| Pages:
|
95-119 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Let $R$ be a noncommutative prime ring of characteristic different from $2$ with Utumi quotient ring $U$ and extended centroid $C$, let $F$, $G$ and $H$ be three generalized derivations of $R$, $I$ an ideal of $R$ and $f(x_1,\ldots ,x_n)$ a multilinear polynomial over $C$ which is not central valued on $R$. If $$F(f(r))G(f(r))=H(f(r)^2)$$ for all $r=(r_1,\ldots ,r_n) \in I^n$, then one of the following conditions holds: \item {(1)} there exist $a\in C$ and $b\in U$ such that $F(x)=ax$, $G(x)=xb$ and $H(x)=xab$ for all $x\in R$; \item {(2)} there exist $a, b\in U$ such that $F(x)=xa$, $G(x)=bx$ and $H(x)=abx$ for all $x\in R$, with $ab\in C$; \item {(3)} there exist $b\in C$ and $a\in U$ such that $F(x)=ax$, $G(x)=bx$ and $H(x)=abx$ for all $x\in R$; \item {(4)} $f(x_1,\ldots ,x_n)^2$ is central valued on $R$ and one of the following conditions holds: \itemitem {(a)} there exist $a,b,p,p'\in U$ such that $F(x)=ax$, $G(x)=xb$ and $H(x)=px+xp'$ for all $x\in R$, with $ab=p+p'$; \itemitem {(b)} there exist $a,b,p,p'\in U$ such that $F(x)=xa$, $G(x)=bx$ and $H(x)=px+xp'$ for all $x\in R$, with $p+p'=ab\in C$. (English) |
| Keyword:
|
prime ring |
| Keyword:
|
derivation |
| Keyword:
|
generalized derivation |
| Keyword:
|
extended centroid |
| Keyword:
|
Utumi quotient ring |
| MSC:
|
16N60 |
| MSC:
|
16W25 |
| idZBL:
|
Zbl 06861569 |
| idMR:
|
MR3783587 |
| DOI:
|
10.21136/CMJ.2017.0352-16 |
| . |
| Date available:
|
2018-03-19T10:26:03Z |
| Last updated:
|
2020-07-06 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/147123 |
| . |
| Reference:
|
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| . |
