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 $\\operatorname{deg} \\tau^{2} \\neq 2$.