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Bol loops with a large left nucleus. (English) Zbl 1192.20051

Starting with an Abelian group \(N\) and a pair of suitable bijections, a power associative loop \(L\) is presented such that the support of \(N\) is a subloop of index 2 contained in the left nucleus of \(L\); a criterion is given when \(L\) is a group. The construction plays the key role in what follows. Possession of a unique nonidentity commutator/associator in a loop is discussed in connection with loop rings. Non-associative Bol loops whose rings in characteristic 2 are strongly right alternative (SRAR loops) are characterized, investigated, and interesting examples are given.

MSC:

20N05 Loops, quasigroups
17D15 Right alternative rings
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