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Alternative loop rings. (English) Zbl 0537.17006

Let R be a commutative and associative ring without elements of additive order 2. Let L be a loop. The following conditions are equivalent: (i) The loop ring RL is alternative. (ii) If x,y,\(z\in L\) are such that \(x.yz=xy.z\) then \(a.bc=ab.c\) whenever \(\{a,b,c\}=\{x,y,z\}\) and if \(x.yz\neq xy.z\) then \(xy.z=x.zy=y.xz.\) (iii) L is an extra loop (i.e. L satisfies the identity \((uv.w)u=u(v.wu)),\quad((x,y,z),x)=1\) for all x,y,\(z\in L\) and if a,b,\(c\in L\) are such that \(a.bc\neq ab.c\) then \((a,b,c)=(a,b)=(b,c)\) (here (d,e,f) and (d,e) denote the associator and the commutator, resp.).
Reviewer: T.Kepka

MSC:

17D05 Alternative rings
20N05 Loops, quasigroups
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