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On derivations and commutativity in prime rings. (English) Zbl 0822.16033
Let $$R$$ be a prime ring, $$U$$ be a right ideal of $$R$$, and $$d$$ be a nonzero derivation of $$R$$. It is shown that each of the following three conditions (i) $$[d(x),d(y)] = d([y,x])$$ for all $$x,y\in R$$, (ii) $$[d(x),d(y)] = d([x,y])$$ for all $$x,y\in R$$, (iii) $$\text{char\,}R\neq 2$$ and $$d([x,y]) = 0$$ for all $$x,y\in R$$, implies that either $$R$$ is commutative or $$d^ 2(U) = d(U)^ 2 = \{0\}$$.

##### MSC:
 16W25 Derivations, actions of Lie algebras 16N60 Prime and semiprime associative rings 16U70 Center, normalizer (invariant elements) (associative rings and algebras) 16U80 Generalizations of commutativity (associative rings and algebras)
##### Keywords:
commutativity theorems; prime rings; right ideals; derivations
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##### References:
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