The concept of derivations as well as generalized derivations (i.e. Ia,b(x) = ax + xb, for all a,b R) have been generalized as an additive function F : R R satisfying F(xy) = F(x)y + xd(y) for all x,y R, where d is a nonzero derivation on R. Such a function F is said to be a generalized derivation. In the present paper it is shown that: if R is 2-torsion free prime ring, I 0 an ideal of R and F a generalized derivation of R such that either F(xy) = F(x)F(y) or F(xy) = F(y)F(x) for all x,y I, then R is commutative

Nadeem-úr Rehman

The concept of derivations as well as generalized derivations (i.e. Ia;b(x)  = ax + xb, for all a; b 2 R) have been generalized as an additive function F: R ! R satisfying F (xy)  = F (x)y + xd(y) for all x; y 2 R, where d is a nonzero derivation on R. Such a function F is said to be a generalized derivation. In the present paper it is shown that: if R is 2-torsion free prime ring, I 6 = 0 an ideal of R and F a generalized derivation of R such that either F (xy)  = F (x)F (y) or F (xy)  = F (y)F (x) for all x; y 2 I, then R is commutative

Nadeem ur Rehman

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ON GENERALIZED DERIVATIONS AS HOMOMORPHISMS AND ANTI-HOMOMORPHISMS

Rehman, Nadeem-úr

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GLASNIK MATEMATICKIVol. 39(59)(2004), 27 { 30ON GENERALIZED DERIVATIONS AS HOMOMORPHISMSAND ANTI-HOMOMORPHISMSNadeem-ur-RehmanBirla Institute of Technology and Science, IndiaAbstract. The concept of derivations as well as generalized deriva-tions (i.e. Ia;b(x) = ax + xb, for all a; b 2 R) have been generalized asan additive function F : R  ! R satisfying F (xy) = F (x)y + xd(y) forall x; y 2 R, where d is a nonzero derivation on R. Such a function F issaid to be a generalized derivation. In the present paper it is shown that:if R is 2-torsion free prime ring, I 6= 0 an ideal of R and F a generalizedderivation of R such that either F (xy) = F (x)F (y) or F (xy) = F (y)F (x)for all x; y 2 I, then R is commutative.1. IntroductionThroughout the present paper R will denote an associative ring withcenter Z(R). For any x; y 2 R, the symbol [x; y] stands for the commutatorxy   yx. Recall that a ring R is called prime if for any a; b 2 R, aRb = (0)implies that either a = 0 or b = 0. An additive mapping d : R  ! R is calleda derivation if d(xy) = d(x)y+xd(y) holds for all x; y 2 R. For a xed a 2 Rthe mapping Ia : R  ! R given by Ia(x) = [x; a] is a derivation which is saidto be inner derivation.An additive function Fa;b : R  ! R is called generalized inner deriva-tion if Fa;b(x) = ax + xb for some xed a; b 2 R. It is straight forward tonote that if Fa;b is a generalized inner derivation, then for any x; y 2 R,Fa;b(xy) = Fa;b(x)y + xIb(y) where Ib is an inner derivation. In view of theabove observation, the concept of generalized derivation is introduced as fol-lows: an additive mapping Fa;b : R  ! R is called a generalized derivationassociated with a derivation d if F (xy) = F (x)y + xd(y) for all x; y 2 R.2000 Mathematics Subject Classication. 16W25, 16N60, 16U80.Key words and phrases. Prime rings, generalized derivations, torsion free rings, homo-morphisms and anti-homomorphisms.2728 NADEEM-UR-REHMANGenerally we do not mention the derivation d associated with a generalizedderivation F rather prefer to call simply a generalized derivation. One mayobserve that the concept of generalized derivation includes the concept ofderivation and generalized inner derivations, also of the left multipliers whend = 0. Hence it should be interesting to extend some results concerningto these notations to generalized derivations. Recently some authors havealso studied generalized derivation in the theory of operator algebras andC   algebras ( see for example [5]).Let S be a nonempty subset of R and F be a generalized derivation of R.If F (xy) = F (x)F (y) or F (xy) = F (y)F (x) for all x; y 2 S, then F is called ageneralized derivation which acts as homomorphisms or anti-homomorphisms,respectively.In [2] Bell and Kappe proved that if a derivation d of a prime ring Rwhich acts as homomorphisms or anti-homomorphisms on a nonzero rightideal of R then d = 0 on R. Further Ashraf et al [1] obtained this result for(; )-derivation. In the present paper our objective is to extend this resultfor generalized derivation acting on ideals in prime ring.Throughout the present paper we shall make extensive use of the followingbasic commutator identities without any specic mention:[xy; z] = x[y; z] + [x; z]y; and [x; yz] = y[x; z] + [x; y]zThe proof of the following lemma can be found in [6].Lemma 1.1. If a prime ring R contains a nonzero commutative rightideal, then R is commutative.Theorem 1.2. Let R be a 2-torsion free prime ring and I be a nonzeroideal of R. Suppose F : R  ! R is a nonzero generalized derivation with d(i) If F acts as a homomorphism on I and if d 6= 0, then R is commutative.(ii) If F acts as an anti-homomorphism on I and if d 6= 0, then R iscommutative.Proof. (i) If F acts as a homomorphism on I , then we have(1) F (xy) = F (x)y + xd(y) = F (x)F (y) for all x; y 2 I:For any x; y; z 2 I , we nd that(2) F (xyz) = F (xy)z + xyd(z) for all x; y; z 2 I:On the other hand(3) F (xyz) = F (x)F (yz) = F (x)F (y)z + F (x)yd(z) for all x; y; z 2 I:On comparing (2) and (3), we get (F (x)  x)yd(z) = 0 for all x; y; z 2 I , andhence (F (x)   x)Id(z) = (0) for all x; z 2 I . Thus, primeness of R forcesthat either (F (x)   x) = 0 or d(z) = 0. If d(z) = 0 for all z 2 I , thend = 0, a contradiction. On the other hand if F (x) = x for all x 2 I , thenON GENERALIZED DERIVATIONS 29xy = F (xy) = F (x)y+xd(y) for all x; y 2 I and hence we nd that xd(y) = 0i.e. Id(y) = 0. Again since I 6= 0, and R is prime we get d(y) = 0 for all y 2 Iand hence d = 0, again a contradiction.(ii) If F acts as an anti-homomorphism(4) F (xy) = F (x)y + xd(y) = F (y)F (x) for all x; y 2 I:Replacing x by xy in (4) and using (4), we get(5) xyd(y) = F (y)xd(y); for all x; y 2 I:Now, replace x by zx in (5), to get(6) zxyd(y) = F (y)zxd(y); for all x; y; z 2 I:Left multiplying (5) by z, we obtain(7) zxyd(y) = zF (y)xd(y); for all x; y; z 2 I:Comparing (6) and (7), we nd that [F (y); z]xd(y) = 0, for all x; y; z 2 Ii.e. [F (y); z]Id(y) = (0). Thus for each y 2 I , by primeness of R either[F (y); z] = 0 or d(y) = 0. Now, let A = fy 2 I j [F (y); z] = 0; for all z 2 Ig,B = fy 2 I j d(y) = 0g. Thus A and B are additive subgroups of I andI = A [ B. But a group can not be a union of two proper subgroups andhence I = A or I = B. If I = B then d(y) = 0 for all y 2 I and henced = 0, a contradiction. On the other hand, if I = A, then [F (y); z] = 0, forall y; z 2 I . Now, replace y by yz to get [y; z]d(z) + y[d(z); z] = 0. Againreplacing y by xy we get [x; z]yd(z) = 0 for all x; y; z 2 I i.e. [x; z]Id(z) = (0).Thus primeness R implies that for each z 2 I either [x; z] = 0 or d(z) = 0.Hence again applying Brauer's trick, we nd that either d(z) = 0 for all z 2 Ior [x; z] = 0, for all x; z 2 I . If d(z) = 0 for all z 2 I then d = 0. Now, if[x; z] = 0 for all x; z 2 I then by Lemma 1.1 we get the required result. Thiscompletes the proof of the theorem.Acknowledgements.The author is greatly indebted to the referee for several useful suggestionsand valuable comments which led to improvement the exposition.References[1] M. Ashraf, N. Rehman and M.A. Quadri, On (; )-derivations in certain classes ofrings, Rad. Math. 9 (1999), 187-192.[2] H.E. Bell and L.C. Kappe, Rings in which derivations satisfy certain algebric condi-tions, Acta Math. Hung. 53 (1989), 339-346.[3] H.E. Bell and W.S. Martindale, Centralizing mappings of semiprime rings, Canad.Math. Bull. 30 (1987), 91-101.[4] I.N. Herstein, Topics in Ring theory, Univ. Chicago press, Chicago 1969.[5] B. Havala, Generalized derivations in rings, Comm. Algebra 26 (1998), 1147-1166.[6] J.H. Mayne, Centralizing mappings of prime rings, Canad. Math. Bull. 27 (1984),122-126.30 NADEEM-UR-REHMAN[7] N. Rehman, On commutativity of rings with generalized derivations, Math. J. OkayamaUniv. 44 (2002), 43-49.[8] M.S. Yenigul and N. Argac, On prime and semi prime rings with -derivations, TurkishJ. Math 18 (1994), 280-284.Department of MathematicsBirla Institute of Technology and SciencePilani 333031Rajasthan, IndiaReceived : 27.09.2002Revised : 15.01.2003

On generalized derivations as homomorphisms and anti-homomorphisms

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