IJPAM: Volume 65, No. 4 (2010)

ON LIE IDEALS AND GENERALIZED
JORDAN $(\theta, \phi)$ DERIVATIONS IN RINGS

Mohammad Asahraf$^1$, Asma Ali$^2$, Muzibur Rahman Mozumder$^3$
$^{1,2,3}$Department of Mathematics
Aligarh Muslim University - AMU
Aligarh, 202002, INDIA
$^1$e-mail: mashraf80@hotmail.com
$^2$e-mail: asma_ali2@rediffmail.com
$^3$e-mail: mrm7862000@yahoo.co.in


Abstract.Let $R$ be a ring and $U$ a Lie ideal of $R$ such that $u^{2} \in U$. Let $ \theta, \phi $ be endomorphisms of $R$ and $M$ be a 2-torsion free $R$-bimodule such that $mRx=\{0\}$ with $ m \in M, x \in R$ implies that either $m=0$ or $x=0$. An additive mapping $F:R \longrightarrow M$ is called a generalized $(\theta, \phi)$ - derivation (resp. generalized Jordon $(\theta, \phi)$-derivation) on $U$ if there exists a $(\theta, \phi)$-derivation $d: R \longrightarrow M$ such that $F(uv) = F(u) \theta(v) + \phi(u)d(v)$ (resp. $F(u^{2}) = F(u) \theta(u) + \phi(u)d(u) )$ holds for all $ u, v \in U$.

In the present paper, it is shown that if $\theta$ is one-one and onto, then every generalized Jordon $(\theta, \phi)$-derivation on $U$ is a generalized $(\theta, \phi)$-derivation on $U$.

Received: May 16, 2010

AMS Subject Classification: 16W25, 16N60, 16U80

Key Words and Phrases: Lie ideals, torsion free rings, derivations, generalized derivations, generalized $(\theta, \phi)$-derivations, generalized Jordan $(\theta, \phi)$-derivations

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2010
Volume: 65
Issue: 4