Geodesic
On left $(\theta,\varphi)$-derivations of prime rings
Archivum mathematicum, Tome 41 (2005) no. 2, pp. 157-166
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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 $R$.
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 $R$.
Classification : 16N60, 16W25
Keywords: Lie ideals; prime rings; derivations; Jordan left derivations; left derivations; torsion free rings 
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Ashraf, Mohammad. On left $(\theta,\varphi)$-derivations of prime rings. Archivum mathematicum, Tome 41 (2005) no. 2, pp. 157-166. http://geodesic.mathdoc.fr/item/ARM_2005_41_2_a3/

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