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On left $(\theta, \phi)$-derivations of prime rings

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*Mohammad Ashraf*

**Address.**

Department of Mathematics,
Faculty of science,
King Abdul Aziz University,
P. O. Box. 80203,
Jeddah 21589, Saudia-Arabia

**E-mail. **mashraf80@hotmail.com

**Abstract.**

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.