%0 Journal Article
%T On generalized left (alpha, beta)-derivations in rings
%J Bulletin of the Iranian Mathematical Society
%I Iranian Mathematical Society (IMS)
%Z 1017-060X
%A Ashraf, Mohammad
%A Ali, Shakir
%A Rehman, Nadeem ur
%A Mozumder, Muzibur Rahman
%D 2012
%\ 12/15/2012
%V 38
%N 4
%P 893-905
%! On generalized left (alpha, beta)-derivations in rings
%K Prime ring
%K Lie ideal
%K Jordan left (alpha,beta)-derivation
%R
%X Let $R$ be a 2-torsion free ring and $U$ be a square closed Lie ideal of $R$. Suppose that $alpha, beta$ are automorphisms of $R$. An additive mapping $delta: R longrightarrow R$ is said to be a Jordan left $(alpha,beta)$-derivation of $R$ if $delta(x^2)=alpha(x)delta(x)+beta(x)delta(x)$ holds for all $xin R$. In this paper it is established that if $R$ admits an additive mapping $G : Rlongrightarrow R$ satisfying $G(u^2)=alpha(u)G(u)+alpha(u)delta(u)$ for all $uin U$ and a Jordan left $(alpha,alpha)$-derivation $delta$; and $U$ has a commutator which is not a left zero divisor, then $G(uv)=alpha(u)G(v)+alpha(v)delta(u)$ for all $u, vin U$. Finally, in the case of prime ring $R$ it is proved that if $G: R longrightarrow R$ is an additive mapping satisfying $G(xy)=alpha(x)G(y)+beta(y)delta(x)$ for all $x,y in R $ and a left $(alpha, beta)$-derivation $delta$ of $R$ such that $G$ also acts as a homomorphism or as an linebreak anti-homomorphism on a nonzero ideal $I$ of $R$, then either $R$ is commutative or $delta=0$ ~on $R$.
%U http://bims.iranjournals.ir/article_296_f014a56e90c11a757243288bb277b4db.pdf