A result on generalized derivations on right ideals of prime rings
Let $R$ be a prime ring of characteristic not 2 and let $I$ be a nonzero right ideal of $R$. Let $U$ be the right Utumi quotient ring of $R$ and let $C$ be the center of $U$. If $G$ is a generalized derivation of $R$ such that $[[G(x), x], G(x)] = 0$ for all $x \in I$, then $R$ is commutative or t...
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| Date: | 2012 |
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| Main Authors: | , , , |
| Format: | Article |
| Language: | English |
| Published: |
Institute of Mathematics, NAS of Ukraine
2012
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| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/2563 |
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| Journal Title: | Ukrains’kyi Matematychnyi Zhurnal |
| Download file: |  |
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Ukrains’kyi Matematychnyi Zhurnal| Summary: | Let $R$ be a prime ring of characteristic not 2 and let $I$ be a nonzero right ideal of $R$.
Let $U$ be the right Utumi quotient ring of $R$ and let $C$ be the center of $U$. If $G$ is a generalized derivation of $R$ such that $[[G(x), x], G(x)] = 0$ for all $x \in I$, then $R$ is commutative or there exist $a, b \in U$ such that $G(x) = ax + xb$ for all $x \in R$ and one of the following assertions is true:
$$(1)\quad (a - \lambda)I = (0) = (b + \lambda)I \;\;\text{for some}\; \lambda \in C,$$
$$(2)\quad (a - \lambda)I = (0) \;\;\text{for some}\; \lambda \in C \;\;\text{and}\; b \in C.$$ |
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