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Alternative loop rings. (English) Zbl 0895.20053

Let \((L,.)\) be a loop, \((R,+,.)\) an associative commutative ring, \(RL\) the loop ring constructed from \(R\) and \(L\). It is clear that \(RL\) is associative iff \(L\) is associative.
The author weakens the associative law to the right alternative law \(xy.y=x.yy\) and the left alternative law \(yy.x=y.yx\) and proves the theorems: – Suppose that \(RL\) satisfies the right alternative law and \(R\) has characteristic \(\neq 2\). Then \(RL\) (and hence also \(L\)) satisfies the left alternative law. – If \(R\) has characteristic 2, then for each \(n\geq 3\) there is a loop \(L\) of size \(2n\) such that \(RL\) is right alternative but \(L\) does not satisfy \(x^3x=xx^3\), which is the special case of the right Bol identity \((xy.z)y=x(yz.y)\) with \(x=y=z\).

MSC:

20N05 Loops, quasigroups
17D05 Alternative rings
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References:

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