24 Journal of Advances in Applied & Computational Mathematics, 2015, 2, 24-29 E-ISSN: 2409-5761/14 © 2014 Avanti Publishers Identities with Generalized Derivations and Automorphisms on Semiprime Rings Asma Ali * , Shahoor Khan and Khalid Ali Hamdin Department of Mathematics Aligarh Muslim University, Aligarh 202002, India Abstract: In this paper we prove some results which extend Theorem 4, Theorem 10 and Theorem 11 of Vukman [13] and proposition 2.3 of Thaheem and Samman [10]. Keywords: Semiprime rings, Derivations, ! -derivations, Generalized derivations. 2010 Mathematics Subject Classification: 16W25, 16R50, 16N60. 1. INTRODUCTION Throughout the paper R will denote an associative ring with centre Z ( R) . Recall that R is prime if for any a,b ! R , aRb = {0} implies that either a = 0 or b = 0 and is semiprime if for any a ! R , aRa = {0} implies that a = 0 . A ring R is said to be a 2 -torsion free if 2x = 0 for x ! R implies that x = 0 . We shall write for any pair of elements x, y ! R the commutator [x, y ] = xy ! yx . We will frequently use the basic commutator identities: [xy, z ] = x[y, z ] + [x, z ] y and [x, yz ] = y[x, z ] + [x, y ] z for all x, y ! R . An additive mapping d : R ! R is called a derivation if d( xy ) = d( x ) y + xd( y ) holds for all x, y ! R . Let ! be an automorphism of a ring R . An additive mapping d : R ! R is called an ! -derivation if d( xy ) = d( x ) !( y ) + xd( y ) holds for all x, y ! R . Note that the mapping d = ! ! I is an ! -derivation. Of course, the concept of ! -derivation generalizes the concept of derivation, since I -derivation is a derivation. An additive mapping F : R ! R is called a generalized derivation with an associated derivation d of R if F( xy ) = F( x ) y + xd( y ) holds for all x, y ! R . Every derivation is a generalized derivation of R . A mapping f : R ! R is called centralizing if [ f ( x ), x ] ! Z ( R) holds for all x ! R , in the special case when [ f ( x ), x ] = 0 holds for all x ! R , the mapping f is said to be commuting on R . Analogously a mapping f : R ! R is called skew-centralizing if f ( x ) x + xf ( x ) ! Z ( R) and is called skew-commuting if f ( x ) x + xf ( x ) = 0 holds for all x ! R . Posner [9] has proved that the existence of nonzero centralizing derivation on a prime ring forces the ring to be commuttative. Mayne [8] proved that in case there *Address correspondence to this author at the Department of Mathematics Aligarh Muslim University, Aligarh 202002, India; E-mail: asma_ali2@rediffmail.com exists a nontrivial centralizing automorphism on a prime ring, then the ring is commuttative. Bresar [2] has proved that if R is a 2-torsion free semiprime ring and f : R ! R is an additive skew- commuting mapping on R , then f = 0 . Vukman [13] proved that if there exist a derivation d : R ! R and an automorphism ! : R ! R , where R is 2-torsion free semiprime ring such that [ d( x ) x + x !( x ), x ] = 0 holds for all x ! R , then d and ! ! I , where I denotes the identity mapping, map R into its center. We extend Vukman results for generalized derivation. 2. MAIN RESULTS We begin with the following Lemmas which are essential to prove our main results. Lemma 2.1. [12, Lemma 1] Let R be a semiprime ring. Suppose that the relation axb + bxc = 0 holds for all x ! R and some a,b,c ! R . In this case, ( a + c) xb = 0 is satisfied for all x ! R . Lemma 2.2. [14, Lemma 1.3] Let R be a semiprime ring. Suppose that there exists a ! R such that a[x, y ] = 0 holds for all x, y ! R . In this case, a ! Z ( R) . Lemma 2.3. [10, Proposition 2.3] Let R be a semiprime ring and let d : R ! R be a commuting ! -derivation on R . In this case, d maps R into its center. Lemma 2.4. [13, Theorem 6] Let R be 2 -torsion free semiprime ring and let f : R ! R be an additive centralizing mappings on R . In this case, f is commuting on R . Lemma 2.5. [13, Lemma 3] Let R be a semiprime ring and let f : R ! R be an additive mapping. If either f ( x ) x = 0 or xf ( x ) = 0 holds for all x ! R , then f = 0 .