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ON α-DERIVATIONS OF PRIME AND SEMIPRIME RINGS

ON α-DERIVATIONS OF PRIME AND SEMIPRIME RINGS DEMONSTRATIO MATHEMATICAVol. XXXVIIINo 22005Joso VukmanON a-DERIVATIONS OF PRIME A N D SEMIPRIME RINGSAbstract. In this paper we investigate identities with a-derivations on prime andsemiprime rings. We prove, for example, the following result. If D : R —* R is ana-derivation of a 2 and 3—torsion free semiprime ring R such that [D(x),x2] = 0 holds,for all x € R, then D maps R into its center. The results of this paper are motivated bythe work of Thaheem and Samman [20].IntroductionThroughout, R is an associative ring with center Z(R). Given an integern > 2, a ring R is said to be n—torsion free if for x € R, nx = 0 impliesx = 0. As usual we write [x, y] for xy — yx and make use of the commutatoridentities [xy, z] — [x, z]y + x[y, z], [x, yz] — [a;, y]z + y[x, z\. We denote by Ithe identity mapping of a ring R. Recall that a ring R is prime if for a,b € R,aRb = (0) implies that either a = 0 or b = 0, and is semiprime in caseaRa = (0) implies a = 0. For explanation of the extended centroid C(R) of asemiprime http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Demonstratio Mathematica de Gruyter

ON α-DERIVATIONS OF PRIME AND SEMIPRIME RINGS

Vukman, Joso
Demonstratio Mathematica , Volume 38 (2): 8 – Apr 1, 2005
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References (10)

Publisher
de Gruyter
Copyright
© by Joso Vukman
ISSN
0420-1213
eISSN
2391-4661
DOI
10.1515/dema-2005-0204
Publisher site
See Article on Publisher Site

Abstract

DEMONSTRATIO MATHEMATICAVol. XXXVIIINo 22005Joso VukmanON a-DERIVATIONS OF PRIME A N D SEMIPRIME RINGSAbstract. In this paper we investigate identities with a-derivations on prime andsemiprime rings. We prove, for example, the following result. If D : R —* R is ana-derivation of a 2 and 3—torsion free semiprime ring R such that [D(x),x2] = 0 holds,for all x € R, then D maps R into its center. The results of this paper are motivated bythe work of Thaheem and Samman [20].IntroductionThroughout, R is an associative ring with center Z(R). Given an integern > 2, a ring R is said to be n—torsion free if for x € R, nx = 0 impliesx = 0. As usual we write [x, y] for xy — yx and make use of the commutatoridentities [xy, z] — [x, z]y + x[y, z], [x, yz] — [a;, y]z + y[x, z\. We denote by Ithe identity mapping of a ring R. Recall that a ring R is prime if for a,b € R,aRb = (0) implies that either a = 0 or b = 0, and is semiprime in caseaRa = (0) implies a = 0. For explanation of the extended centroid C(R) of asemiprime

Journal

Demonstratio Mathematica – de Gruyter

Published: Apr 1, 2005

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