International Journal of Algebra, Vol. 1, 2007, no. 11, 547 - 550 An Identity Related to Generalized Derivations M. N. Daif Department of Mathematics, Faculty of Science Al-Azhar University, Nasr City (11884), Cairo, Egypt nagydaif@yahoo.com M. S. Tammam El-Sayiad Department of Mathematics, Faculty of Science Beni Suef University, Beni Suef (62111), Egypt m s tammam@yahoo.com Abstract Let R be a 2−torsion free semiprime ring such that R has a commu- tator which is not a zero divisor and G: R−→R be an additive mapping such that G(xyx)= G(x)yx + xD(yx) holds for all x, y ∈ R for some derivation D. Then G is a generalized derivation. Mathematics Subject Classification: 16W10, 16E99 Keywords: prime ring, semiprime ring, left(right) centralizer, Jordan left(right) centralizer, derivation, Jordan derivation, generalized derivation, Jordan generalized derivation 1 Introduction This note has been motivated by the work of Moln´ ar [4] and Vukman and Kosi- Ulbl [5]. Throughout, R will represent an associative ring with center Z (R). A ring R is 2-torsion free, if 2x = 0, x ∈ R implies x = 0. Recall that R is prime if aRb = (0) implies a = 0 or b = 0, and semiprime if aRa = (0) implies a = 0. An additive mapping T : R −→ R is called a left (right) centralizer in case T (xy )= T (x)y (T (xy )= xT (y )) holds for all x, y ∈ R. Zalar [6] has proved that any left (right) Jordan centralizer on a 2-torsion free semiprime ring is a left (right) centralizer. Moln´ ar [4] has proved the following result: Let R be a 2-torsion free prime ring and let T : R −→ R be an additive mapping.