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.


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


Zbl 0696.17016
Full Text: DOI


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