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IDENTITIES WITH PRODUCTS OF (α,β)-DERIVATIONS ON PRIME RINGS

IDENTITIES WITH PRODUCTS OF (α,β)-DERIVATIONS ON PRIME RINGS DEMONSTRATE MATHEMATICAVol. XXXIXNo 22006Joso VukmanIDENTITIES WITH PRODUCTS OF (<*,/?)-DERIVATIONSON PRIME RINGSAbstract. The main purpose of this paper is to prove the following result. Let R bea noncommutative prime ring of characteristic different from two and let D and G / 0 be(a, /3)-derivations of R into itself such that G commutes with a and ¡3. If [D(x), G(x)] = 0holds for all x € R then D = AG where A is an element from the extended centroid of R.This research has been motivated by the work of Chaudhry, Sammanand Thaheem [9,10,11,15] and is a continuation of our earlier work [16].Throughout, R is an associative ring with center Z(R). As usual we write[a;, y] for xy—yx and make use of the commutator identities [xy, z] = [x, z]y+x\y, z], [x, yz] = [x, y]z + y[x, z]. We denote by I the identity mapping of aring R. Recall that a ring R is prime if for a,b G R, aRb = (0) implies thateither a = 0 or b = 0, and is semiprime in case aRa = (0) implies thata = 0. For explanation of the extended centroid C(R) of a semiprime ringR we refer to [1]. An http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Demonstratio Mathematica de Gruyter

IDENTITIES WITH PRODUCTS OF (α,β)-DERIVATIONS ON PRIME RINGS

Demonstratio Mathematica , Volume 39 (2): 8 – Apr 1, 2006

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Publisher
de Gruyter
Copyright
© by Joso Vukman
ISSN
0420-1213
eISSN
2391-4661
DOI
10.1515/dema-2006-0207
Publisher site
See Article on Publisher Site

Abstract

DEMONSTRATE MATHEMATICAVol. XXXIXNo 22006Joso VukmanIDENTITIES WITH PRODUCTS OF (<*,/?)-DERIVATIONSON PRIME RINGSAbstract. The main purpose of this paper is to prove the following result. Let R bea noncommutative prime ring of characteristic different from two and let D and G / 0 be(a, /3)-derivations of R into itself such that G commutes with a and ¡3. If [D(x), G(x)] = 0holds for all x € R then D = AG where A is an element from the extended centroid of R.This research has been motivated by the work of Chaudhry, Sammanand Thaheem [9,10,11,15] and is a continuation of our earlier work [16].Throughout, R is an associative ring with center Z(R). As usual we write[a;, y] for xy—yx and make use of the commutator identities [xy, z] = [x, z]y+x\y, z], [x, yz] = [x, y]z + y[x, z]. We denote by I the identity mapping of aring R. Recall that a ring R is prime if for a,b G R, aRb = (0) implies thateither a = 0 or b = 0, and is semiprime in case aRa = (0) implies thata = 0. For explanation of the extended centroid C(R) of a semiprime ringR we refer to [1]. An

Journal

Demonstratio Mathematicade Gruyter

Published: Apr 1, 2006

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