Department of Mathematics, Faculty of science, King Abdul Aziz University, P. O. Box. 80203, Jeddah 21589, Saudia-Arabia
Let $R$ be a $2$-torsion free prime ring. Suppose that $\theta, \phi$ are automorphisms of $R$. In the present paper it is established that if $R$ admits a nonzero Jordan left $(\theta,\theta)$-derivation, then $R$ is commutative. Further, as an application of this resul it is shown that every Jordan left $(\theta,\theta)$-derivation on $R$ is a left $(\theta,\theta)$-derivation on $R$. Finally, in case of an arbitrary prime ring it is proved that if $R$ admits a left $(\theta,\phi)$-derivation which acts also as a homomorphism (resp.\ anti-homomorphism) on a nonzero ideal of $R$, then $d=0$ on $R$.
AMSclassification. 16W25, 16N60.
Keywords. Lie ideals, prime rings, derivations, Jordan left derivations, left derivations, torsion free rings.