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On orthogonal generalized derivations of semiprime rings

In this paper, we present some results concerning two generalized derivations on a semiprime ring. These results are a generalization of results of M. Bresar and J. Vukman, which are related to a theorem of E. Posner for the product of derivations on a prime ring. | Turk J Math 28 (2004) , 185 – 194. ¨ ITAK ˙ c TUB On Orthogonal Generalized Derivations of Semiprime Rings Nurcan Arga¸c, Atsushi Nakajima and Emine Alba¸s Abstract In this paper, we present some results concerning two generalized derivations on a semiprime ring. These results are a generalization of results of M. Bre˘sar and J. Vukman in [2], which are related to a theorem of E. Posner for the product of derivations on a prime ring. Key words and phrases: derivation, orthogonal derivations, generalized derivation, orthogonal generalized derivations, prime ring, semiprime ring. 1. Introduction Throughout R will represent an associative ring. R is said to be 2-torsion free if 2x = 0, x ∈ R implies x = 0. Recall that R is prime if xRy = 0 implies x = 0 or y = 0, and R is semiprime if xRx = 0 implies x = 0. An additive mapping d : R → R is called a derivation if d(xy) = d(x)y + xd(y) holds for all x, y ∈ R. In [1], Bre˘sar defined the following notion. An additive mapping D : R → R is said to be a generalized derivation if there exists a derivation d : R → R such that D(xy) = D(x)y + xd(y) for all x, y ∈ R. 1991 AMS Mathematics Subject Classification: 16W25 This paper is dedicated to the memory of Professor Mehmet Sapanci 185 ARGAC ¸ , NAKAJIMA, ALBAS ¸ Hence the concept of a generalized derivation covers both the concepts of a derivation and of a left multiplier ( i.e., an additive map f satisfying f(xy) = f(x)y for all x, y ∈ R). This notion is found in P. Ribenboim [8], where some module structure of these higher generalized derivations was treated. Other properties of generalized derivations were given by B. Hvala [3], T.K. Lee [4] and the second author [5], [6] and [7]. We note that for a semiprime ring R, if D is a function from R to R and d : R → R is an additive mapping such that D(xy) = D(x)y+xd(y) for all x, y ∈ R, then D is uniquely determined by d and moreover d must be a derivation by [[1], Remark 1]. Let d : R → R be a derivation and a an .