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On Jordan left derivations. (English) Zbl 0813.16021

Let \(R\) be a ring and \(X\) be a left \(R\)-module such that \(aRx = 0\), where \(a \in R\) and \(x \in X\), implies \(a = 0\) or \(x = 0\). Suppose there exists a nonzero additive map \(D : R \to X\) satisfying \(D(a^ 2) = 2aD(a)\) for every \(a \in R\) (such maps are called Jordan left derivations). J. Vukman and the reviewer [Proc. Am. Math. Soc. 110, 7-16 (1990; Zbl 0703.16020)] have shown that if the characteristic of \(R\) is different from 2 and 3, then \(R\) must be commutative. In the paper under review, the condition that the characteristic is different from 3 is removed.

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)

Citations:

Zbl 0703.16020
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