Loops whose loop rings are alternative. (English) Zbl 0582.17015

Let R be a non-trivial commutative and associative ring with identity such that 2a\(\neq 0\) for every \(0\neq a\in R\). A loop L is said to be an RA-loop if the loop ring RL of L over R is alternative but not associative. In this case, L is a Moufang loop, the nucleus N(L) and the centre C(L) of L coincide, \(a^ 2\in N(L)\) for every \(a\in L\) and, for all a,b\(\in L\), \((a,b)=1\) iff \((a,b,c)=1\) for any \(c\in L\). Moreover, the associator and the commutator subloop of L are equal subgroups of order 2 of C(L) and the loop L has the property LC (i.e., if a,b\(\in L\) and \(ab=ba\) then \(\{a,b,ab\}\cap C(L)\neq \emptyset).\) Conversely, a non- associative Moufang loop is an RA-loop iff it has LC and a unique non- identity commutator.
Reviewer: T.Kepka


17D05 Alternative rings
20N05 Loops, quasigroups
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