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On derivations and commutativity of prime rings with involution

On derivations and commutativity of prime rings with involution Abstract In (Acta Math. Hungar. 66 (1995), 337–343), Bell and Daif proved that if R is a prime ring admitting a nonzero derivation such that d ( x y ) = d ( y x ) ${d(xy)=d(yx)}$ for all x , y ∈ R ${x,y\in R}$ , then R is commutative. The objective of this paper is to examine similar problems when the ring R is equipped with involution. It is shown that if a prime ring R with involution * of a characteristic different from 2 admits a nonzero derivation d such that d ( x x * ) = d ( x * x ) ${d(xx^*)=d(x^*x)}$ for all x ∈ R and S ( R ) ∩ Z ( R ) ≠ ( 0 ) ${S(R)\cap Z(R)\ne (0)}$ , then R is commutative. Moreover, some related results have also been discussed. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Georgian Mathematical Journal de Gruyter

On derivations and commutativity of prime rings with involution

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References (15)

Publisher
de Gruyter
Copyright
Copyright © 2016 by the
ISSN
1072-947X
eISSN
1572-9176
DOI
10.1515/gmj-2015-0016
Publisher site
See Article on Publisher Site

Abstract

Abstract In (Acta Math. Hungar. 66 (1995), 337–343), Bell and Daif proved that if R is a prime ring admitting a nonzero derivation such that d ( x y ) = d ( y x ) ${d(xy)=d(yx)}$ for all x , y ∈ R ${x,y\in R}$ , then R is commutative. The objective of this paper is to examine similar problems when the ring R is equipped with involution. It is shown that if a prime ring R with involution * of a characteristic different from 2 admits a nonzero derivation d such that d ( x x * ) = d ( x * x ) ${d(xx^*)=d(x^*x)}$ for all x ∈ R and S ( R ) ∩ Z ( R ) ≠ ( 0 ) ${S(R)\cap Z(R)\ne (0)}$ , then R is commutative. Moreover, some related results have also been discussed.

Journal

Georgian Mathematical Journalde Gruyter

Published: Mar 1, 2016

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