Miskolc Mathematical Notes HU e-ISSN 1787-2413 Vol. 15 (2014), No. 2, pp. 717–724 ON IDEALS WITH SKEW DERIVATIONS OF PRIME RINGS NADEEM UR REHMAN AND MOHD ARIF RAZA Received 24 April, 2014 Abstract. Let R be a prime ring and set Œx;y 1 D Œx;y D xy  yx for all x;y 2 R and induct- ively Œx;y k D ŒŒx;y k1 ;y for k>1. We apply the theory of generalized polynomial identities with automorphism and skew derivations to obtain the following result: Let R be a prime ring and I a nonzero ideal of R. Suppose that .ı;'/ is a skew derivation of R such that ı.Œx;y/ D Œx;y n for all x;y 2 I , then R is commutative. 2010 Mathematics Subject Classification: 16N20; 16W25; 16N55; 16N60 Keywords: skew derivation, automorphism, generalized polynomial identity (GPI), prime ring, ideal 1. I NTRODUCTION,NOTATION AND STATEMENTS OF THE RESULTS Throughout this paper, unless specifically stated, R is always an associative prime ring with center Z.R/, Q its Martindale quotient ring. Note that Q is also prime and the center C of Q, which is called the extended centroid of R, is field (we refer the reader to [1] for the definitions and related properties of these objects). For any x;y 2 R, the symbol Œx;y stands for the commutator xy  yx. Recall that a ring R is called prime if for any x;y 2 R, xRy Df0g implies that either x D 0 or y D 0. An additive mapping d W R ! R is called a derivation if d.xy/ D d.x/y C xd.y/ holds for all x;y 2 R. An additive mapping F W R ! R is called a generalized derivation if there exists a derivation d W R ! R such that F.xy/ D F.x/y C xd.y/ holds for all x;y 2 R, denoted by .F;d/. Hence, the concept of generalized derivations covers both the concepts of a derivation and of a left multiplier. Given any automorphism ' of R, an additive mapping ı W R ! R satisfying ı.xy/ D ı.x/y C '.x/ı.y/ for all x;y 2 R is called a ' -derivation of R, or a skew de- rivation of R with respect to ' , denoted by .ı;'/. It is easy to see if ' D 1 R , the iden- tity map of R, then a ' -derivation is merely an ordinary derivation. And if ' ¤ 1 R , then '  1 R is a skew derivation. Thus the concept of skew derivations can be regard as a generalization of both derivations and automorphism. When ı.x/ D '.x/b  bx for some b 2 Q, then .ı;'/ is called an inner skew derivation, and otherwise it is outer. Any skew derivation .ı;'/ extends uniquely to a skew derivation of Q [12] via extensions of each map to Q. Thus we may assume that any skew derivation of c  2014 Miskolc University Press