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A Non-Abelian Near Ring in Which (-1)r=r Implies r=0

Published online by Cambridge University Press:  20 November 2018

Bruce McQuarrie
Bruce McQuarrie*
Affiliation:
Worcester Polytechnic Institute, Worcester, Massachusetts 01609, U.S.A.
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In this Bulletin Ligh [2] generalized to finite near rings with identity a theorem Zassenhaus [5] used to prove every finite near field has abelian addition. B. H. Neumann [4] extended Zassenhaus’ result, using similar techniques and showed that all near fields are abelian. It has been an open question whether Ligh’s generalization could be carried out to infinite near rings with identity. The purpose of this paper is to show that Ligh’s theorem cannot be so extended. In particular, it cannot be extended even to distributively generated near rings, a type of near ring which has been useful in studying endomorphism rings of non-abelian groups [1,3].

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 17 , Issue 1 , 01 March 1974 , pp. 73 - 75
Copyright
Copyright © Canadian Mathematical Society 1974

References

1. Frolich, A., Distributively generated near rings. Proc. London Math. Soc. 8 (1958), 76-94.Google Scholar
2. Ligh, S., A generalization of a theorem ofZassenhaus. Can. Math. Bull. 12 (1969), 677-678.Google Scholar
3. McQuarrie, B. and Malone, J. J., Endomorphism rings of non-abelian groups. Bull. Austral. Math. Soc. 3 (1970), 349-352.Google Scholar
4. Neumann, B. H., On the commutativity of addition. J. London Math. Soc. 15 (1940), 203-208.Google Scholar
5. H. Zassenhaus, Uber endlich Fastkorper9 Abh. Math. Sem. Univ. Hamburg 11 (1936), 187-220.Google Scholar