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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 \(d^ 2(U) = d(U)^ 2 = \{0\}\).

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)
Full Text: DOI
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