## 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:

  Chein O., Algebras Groups Geom 5 pp 297– (1988)  Goodaire E.G., Resenhas 2 pp 47– (1995)  DOI: 10.1080/00927879408825150 · Zbl 0821.20060  Goodaire E.G., Results in Mathematics 29 pp 56– (1996)  DOI: 10.1007/BF00881714 · Zbl 0828.68109  DOI: 10.2172/10129052  Zhang, J. and Zhang, H. SEM: a system for enumerating models. Proc. 14th Int. Joint Conf. on AI (IJCAI-95). pp.298–303. Montréal Available on WWW at: http://www.cs.uiowa.edu/ hzhang/
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