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A construction of loops which admit right alternative loop rings. (English) Zbl 0861.20058

It is known [see O. Chein and E. G. Goodaire, Algebras Groups Geom. 5, No. 3, 297-304 (1989; Zbl 0696.17016)] that – if a loop ring \(RL\) is right alternative and \(\text{char }R\neq 2\), then \(RL\) is necessarily left alternative, – if \(L\) is Moufang and if \(RL\) is right alternative (regardless of the characteristic of \(R\)), then \(RL\) must be left alternative. Thus if \(RL\) is a right, but not left, alternative ring, then \(\text{char }R\) must be 2 and \(L\) cannot be Moufang. Theorem 3.1. of this paper gives a specific construction of loops whose loop rings are right, but not left, alternative.

MSC:

20N05 Loops, quasigroups
17D15 Right alternative rings
17D05 Alternative rings

Citations:

Zbl 0696.17016
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Full Text: DOI

References:

[1] R. H. Bruck, A survey of binary systems, Ergeb. Math. Grenzgeb., vol. 20, Springer-Verlag, 1958. · Zbl 0081.01704
[2] Orin Chein and Edgar G. Goodaire, Is a right alternative loop ring alternative?, Algebras Groups Geom. 5 (1988), 297–304. · Zbl 0696.17016
[3] Edgar G. Goodaire and D. A. Robinson, A class of loops whose loop rings are strongly right alternative, Comm. Algebra 22 (1994), no. 14, 5623–5634. · Zbl 0821.20060 · doi:10.1080/00927879408825150
[4] A. Kreuzer and H. Wefelscheid, On k-loops of finite order, Resultate Math. 25 (1994), no. 1-2, 79–102. · Zbl 0803.20052 · doi:10.1007/BF03323144
[5] D. A. Robinson, Bol loops, Trans. Amer. Math. Soc. 123 (1966), 341–354. · Zbl 0163.02001 · doi:10.1090/S0002-9947-1966-0194545-4
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