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On derivations and commutativity in semiprime rings. (English) Zbl 0832.16033

Let \(R\) be a semiprime ring with center \(Z\), \(d\) be a derivation of \(R\) and \(U\) be a nonzero left ideal of \(R\) such that \(d(U)\neq 0\). It is shown that there exists a nonzero central ideal of \(R\) if one of the following two conditions is fulfilled: (i) \(R\) is 6-torsion-free and \([[d(x),x],x]\in Z\) for all \(x\in U\); (ii) there is a positive integer \(n\) such that \(R\) is \(n!\)-torsion-free and \([d(x),x^n]\in Z\) for all \(x\in U\).

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)
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[1] DOI: 10.4153/CMB-1987-014-x · Zbl 0614.16026 · doi:10.4153/CMB-1987-014-x
[2] Brešar M., Proc. Amer. Math. Soc 114 pp 641– (1992)
[3] DOI: 10.1090/S0002-9939-1979-0539624-9 · doi:10.1090/S0002-9939-1979-0539624-9
[4] DOI: 10.1090/S0002-9939-1990-1007517-3 · doi:10.1090/S0002-9939-1990-1007517-3
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