Generalized derivations acting on multilinear polynomials as Jordan homomorphisms (Q2680453)

Let \(R\) be a prime ring with its Utumi quotient ring \(U\), the extended centroid \(C\), and \(Z(R)\) the center of \(R\). It may be noted that the extended centroid \(C\) of a prime ring \(R\) is always a field and \(C=Z(U)\). An additive mapping \(d:R \to R\) is said to be a derivation of \(R\) if \(d(xy)=d(x)y+xd(y)\) holds for all \(x,y \in R\). An additive mapping \(H:R \to R\) is called a generalized derivation of \(R\) if there exists a derivation \(d:R\to R\) such that \(H(xy)=H(x)y+xd(y)\) holds for all \(x,y\in R\). An additive mapping \(H\) is said to be a Jordan homomorphism if \(H(x^2)=(H(x))^2\) holds for all \(x\in R\). It is straightforward to see that every homomorphism on a ring \(R\) is a Jordan homomorphism but there exist Jordan homomorphisms on \(R\) which are not a homomorphism on \(R\). It was \textit{I. N. Herstein} [Trans. Am. Math. Soc. 81, 331--341 (1956; Zbl 0073.02202)] who proved that if \(H\) is a Jordan homomorphism of a ring \(R\) onto a prime ring \(R^{\prime}\) of characteristic different from \(2\) and \(3\), then \(H\) is either a homomorphism or an anti-homomorphism. Further, \textit{M. F. Smiley} [Trans. Am. Math. Soc. 84, 426--429 (1957; Zbl 0089.25901)] improved the above result by removing the restriction that \(R^{\prime}\) has characteristic different from \(3\). In the paper under review the authors obtain a complete description of generalized derivations \(F\) and \(H\) when F acts as a Jordan homomorphism on a multilinear polynomial of a prime ring R, that is, it satisfies the property \[ G(H(f(r_{1},r_{2},\dots,r_{n})^{2})) = (H(f(r_{1},r_{2},\dots,r_{n})^{2}))^{2} \] for all \(r_{1},r_{2},\dots,r_{n}\in R\) where \(F\) is a generalized derivation and \(f(r_{1},r_{2},\dots,r_{n})\) is a multilinear polynomial over \(R\). In fact, it is shown that if \(R\) is a non-commutative prime ring of characteristic different from \(2\) and \(F\) and \(H\) are generalized derivations on \(R\), \(f(r_{1},r_{2},\dots,r_{n})^{2}\) is a multilinear polynomial over \(C\) which is not central valued on \(R\), then one of the following holds: \begin{itemize} \item[1.] \(H = 0\); \item[2.] there exists \(\lambda \in C\) such that \(G(x) = H(x) = \lambda x\) for all \(x \in R\); \item[3.] there exist \(\lambda \in C\) and \(a \in U\) such that \(H(x) = \lambda x\), \(G(x)=[a, x] + \lambda x\) for all \(x \in R\), and \(f(r_{1},r_{2},\dots,r_{n})^{2}\) is central-valued on \(R\). \end{itemize}