Geodesic
A Non-Abelian Near Ring in Which (-1)r=r Implies r=0
Canadian mathematical bulletin, Tome 17 (1974) no. 1, pp. 73-75

Voir la notice de l'article provenant de la source Cambridge University Press

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]. 
McQuarrie, Bruce. A Non-Abelian Near Ring in Which (-1)r=r Implies r=0. Canadian mathematical bulletin, Tome 17 (1974) no. 1, pp. 73-75. doi: 10.4153/CMB-1974-012-4
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[1] 1. Frolich, A., Distributively generated near rings. Proc. London Math. Soc. 8 (1958), 76-94. Google Scholar

[2] 2. Ligh, S., A generalization of a theorem ofZassenhaus. Can. Math. Bull. 12 (1969), 677-678. Google Scholar

[3] 3. McQuarrie, B. and Malone, J. J., Endomorphism rings of non-abelian groups. Bull. Austral. Math. Soc. 3 (1970), 349-352. Google Scholar

[4] 4. Neumann, B. H., On the commutativity of addition. J. London Math. Soc. 15 (1940), 203-208. Google Scholar

[5] 5. H. Zassenhaus, Uber endlich Fastkorper9 Abh. Math. Sem. Univ. Hamburg 11 (1936), 187-220. Google Scholar

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