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.


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)


Zbl 0703.16020