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Jordan left derivations on semiprime rings. (English) Zbl 0937.16044
If \(R\) is a ring then \(D\in\text{End}(R,+)\) is a Jordan left derivation of \(R\) when \(D(x^2)=2xD(x)\) for all \(x\in R\). The author proves two results for such maps analogous to results known for derivations. These are: if \(R\) is a 2-torsion free semiprime ring, \(D\) a Jordan left derivation of \(R\), and \(n>1\) so that \(D(x)^n=0\) for all \(x\in R\), then \(D=0\); and if also \(R\) is 3-torsion free then \(D^2\) a Jordan left derivation of \(R\) forces \(D=0\). In the paper the second result is mistakenly stated for Jordan derivations rather than for Jordan left derivations.

16W25 Derivations, actions of Lie algebras
16N60 Prime and semiprime associative rings