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 \(x\in U\).


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


[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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