Eng. & Tech. Journal, Vol.29, No.3, 2011 * Applied Sciences Department, University of Technology /Baghdad ** College of Science, University of Baghdad /Baghdad 544 On Left σ -Centralizers of Jordan Ideals And Generalized Jordan Left ) , ( τ σ -Derivations of Prime Rings Dr. Anwar Khaleel Faraj * & Dr. Abdulrahman H. Majeed** Received on: 6/6/2010 Accepted on: 3/2/2011 Abstract In this paper we generalize the result of S. Ali and C. Heatinger on left σ - centralizer of semiprime ring to Jordan ideal, we proved that if R is a 2-torsion free prime ring, U is a Jordan ideal of R and G is an additive mapping from R into itself satisfying the condition ) ( ) ( ) ( ) ( ) ( u r G r u G ru ur G σ σ + = + , for all R r U u ∈ ∈ , . Then ) ( ) ( ) ( r u G ur G σ = , for all R r U u ∈ ∈ , . Also, we extend the result of S. M. A. Zaidi, M. Ashraf and S. Ali on left ( σ , σ )-derivation of prime ring to Jordan ideal by introducing the concept of generalized Jordan left ( σ , τ )- derivation. Keywords: centralizer, σ -centralizer, ( σ , τ )-derivation, left ( σ , τ )-derivation, generalized ( σ , τ )- derivation, prime ring. زر ول - σ ت وردان و ت ر ا- ) , ( τ σ رىوردان ا وت ا ا ا ث ذا ا ھ S. Ali وC. Heatinger زر - σ ا روردان ! او ا, ت اذارھ R طن ا واء ا ط او 2 , U ! وردان R وG ن % داR اR ث ) ( ) ( ) ( ) ( ) ( u r G r u G ru ur G σ σ + = + , ل u ϶ U , r ϶ R . ً ن) ) ( ) ( ) ( r u G ur G σ = , ل u ϶ U , r ϶ R . كذ و, S. M. A. Zaidi , M. Ashraf وS. Ali )- σ , σ ( وم,- مد وردان ! او ا رى ا- ) , ( τ σ رىوردان ا % ا. 1. Introduction hroughout the present paper R will denote an associative ring with center ) ( R Z , not necessarily with an identity element. We will write for all - = ∈ xy y x R y x ] , [ , , yx and yx xy y x + = for the Lie product and Jordan product, respectively. A ring R is said to be prime if 0 = xRy implies that 0 = x or 0 = y and R is semiprime in case 0 = xRx implies 0 = x , [1]. An additive subgroup U of R is said to be Jordan ideal (resp. Lie ideal) of R if U r u ∈ (resp. R r u ∈ ] , [ ), for all R r U u ∈ ∈ , , [1]. A ring R is called n-torsion free, T