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A NOTE ON THE COMMUTATIVITY OF RINGS

A NOTE ON THE COMMUTATIVITY OF RINGS DEMONSTRATIO MATHEMATICAVol. XXXIXNo 22006Pawel Andrzejewski, Barbara GlancA NOTE ON THE COMMUTATIVITY OF RINGSAbstract. Sufficient conditions for commutativity of rings are proved. They generalize or are related to certain old results due to I. N. Herstein and others, see [1] and [5].1. IntroductionLet P denote an arbitrary (associative) ring (we do not assume that P hasan identity element). For any positive integer n, we consider the followingtwo conditions V{n) and Q(n) imposed on elements of the ring P :V(n):Q(n):A{x-y)nx,yePA(x-y)nx,yeP==xn-yn,(yx)n.Obviously, the conditions V(n) and Q(n) hold in any commutative ringP. Moreover it can be easily verified that if the condition V(2) holds in thering P with an identity element 1 then P is commutative. Similarly, if thering P with an identity element 1 is of characteristic different from two andsatisfies the condition Q(2) then P is also commutative.In the same spirit, one can prove that the ring P satisfying the conditionV{2) or Q(2) is commutative provided that it contains no non-zero nilpotentelements.The purpose of this note is to prove a generalization of the above observations. Among other things, we show that under certain mild assumptions,the ring P is commutative if it satisfies any of the conditions V(n) or Q(n).In the fifties http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Demonstratio Mathematica de Gruyter

A NOTE ON THE COMMUTATIVITY OF RINGS

Andrzejewski, Paweł; Glanc, Barbara
Demonstratio Mathematica , Volume 39 (2): 6 – Apr 1, 2006
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References (4)

Publisher
de Gruyter
Copyright
© by Paweł Andrzejewski
ISSN
0420-1213
eISSN
2391-4661
DOI
10.1515/dema-2006-0208
Publisher site
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Abstract

DEMONSTRATIO MATHEMATICAVol. XXXIXNo 22006Pawel Andrzejewski, Barbara GlancA NOTE ON THE COMMUTATIVITY OF RINGSAbstract. Sufficient conditions for commutativity of rings are proved. They generalize or are related to certain old results due to I. N. Herstein and others, see [1] and [5].1. IntroductionLet P denote an arbitrary (associative) ring (we do not assume that P hasan identity element). For any positive integer n, we consider the followingtwo conditions V{n) and Q(n) imposed on elements of the ring P :V(n):Q(n):A{x-y)nx,yePA(x-y)nx,yeP==xn-yn,(yx)n.Obviously, the conditions V(n) and Q(n) hold in any commutative ringP. Moreover it can be easily verified that if the condition V(2) holds in thering P with an identity element 1 then P is commutative. Similarly, if thering P with an identity element 1 is of characteristic different from two andsatisfies the condition Q(2) then P is also commutative.In the same spirit, one can prove that the ring P satisfying the conditionV{2) or Q(2) is commutative provided that it contains no non-zero nilpotentelements.The purpose of this note is to prove a generalization of the above observations. Among other things, we show that under certain mild assumptions,the ring P is commutative if it satisfies any of the conditions V(n) or Q(n).In the fifties

Journal

Demonstratio Mathematica – de Gruyter

Published: Apr 1, 2006

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