On derivations and commutativity in prime rings. (English) Zbl 1078.16036

The following result is proved: Let \(R\) be a prime ring of characteristic \(\neq 2\) and \(d\) a nonzero derivation in \(R\). Then (i) if \(R\) satisfies the differential identity \([[d(x),x],[d(y),y]]=0\) then \(R\) is commutative; (ii) if a nonzero right ideal \(I\) of \(R\) satisfies the identity \([[d(x),x],[d(y),y]]=0\) then either \([I,I]I=0\) or \(d(I)I=0\).


16W25 Derivations, actions of Lie algebras
16N60 Prime and semiprime associative rings
16R50 Other kinds of identities (generalized polynomial, rational, involution)
16U70 Center, normalizer (invariant elements) (associative rings and algebras)
Full Text: DOI EuDML