Jordan $\tau$-derivations of Prime GPI-rings

Abstract

Let $R$ be a noncommutative prime ring, with maximal symmetric ring of quotients $Q_{ms}(R)$ and extended centriod $C$, and let $\tau$ be an anti-automorphism of $R$. An additive map $\delta \colon R \to Q_{ms}(R)$ is called a Jordan $\tau$-derivation if $\delta(x^2) = \delta(x) x^{\tau} + x\delta(x)$ for all $x \in R$. In 2015 Lee and the author proved that any Jordan $\tau$-derivation of $R$ is X-inner if either $R$ is not a GPI-ring or $R$ is a PI-ring except when $\operatorname{char}R = 2$ and $\dim_C RC = 4$. In the paper we prove that, when $R$ is a prime GPI-ring but is not a PI-ring, any Jordan $\tau$-derivation is X-inner if either $\tau$ is of the second kind or both $\operatorname{char}R \neq 2$ and $\tau$ is of the first kind with $\operatorname{deg} \tau^{2} \neq 2$.