Jordan derivations and Jordan homomorphisms on prime rings of characteristic 2
The main result of this paper shows that if \(T\) is a bijective and additive mapping from a ring \(R\) to a noncommutative prime ring \(S\) of characteristic 2, and if \(T\) satisfies \(T(xy+yx)=T(x)T(y)+T(y)T(x)\) and \(T(x^3)=T(x)^3\) for all \(x,y\in R\), then \(T\) is either an isomorphism or an anti-isomorphism. This theorem extends a similar one of \textit{I. N. Herstein} and \textit{E. Kleinfeld} [Pac. J. Math. 10, 843-852 (1960; Zbl 0093.040)] by making some slight modifications in its proof. The earlier result assumes that \(T\) is onto and that \(R\) and \(S\) are simple rings. The author also asserts that when \(D\) is an additive mapping from the noncommutative prime ring \(R\) of characteristic 2 to itself so that \(D\) acts like a derivation on commutators and on cubes, then \(D\) is a derivation of \(R\). This extends a result of \textit{I. N. Herstein} [Proc. Am. Math. Soc. 8, 1104-1110 (1958; Zbl 0216.07202)] which assumes that \(D\) acts like a derivation on commutators and on products of the form \(xyx\).
scientific article; zbMATH DE number 4042112