A note on generalized derivations as a Jordan homomorphisms

Bull. Korean Math. Soc. Published online February 13, 2020

C. Arusha and S. K. Tiwari
Indian Institute of Science Education and Research Bhopal

Abstract : Let $R$ be a prime ring of characteristic different from $2$. Suppose that $F$, $G$, $H$ and $T$ are generalized derivations of $R$. Let $U$ be the Utumi quotient ring of $R$ and $C$ be the center of $U$, called the extended centroid of $R$ and let $f(x_1,\ldots,x_n)$ be a non central multilinear polynomial over $C$. If
\bea
&F(f(r_1,\ldots,r_n))G(f(r_1,\ldots,r_n))-f(r_1,\ldots,r_n)T(f(r_1,\ldots,r_n))\nn\\
&=H(f(r_1,\ldots,r_n)^2)\nn
\eea

for all $r_1, \ldots, r_n \in R$, then we describe all possible forms of $F$, $G$, $H$ and $T$.