## 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
Full Text:

### References:

 [1] Chein O., Algebras Groups Geom 5 pp 297– (1988) [2] Goodaire E.G., Resenhas 2 pp 47– (1995) [3] DOI: 10.1080/00927879408825150 · Zbl 0821.20060 [4] Goodaire E.G., Results in Mathematics 29 pp 56– (1996) [5] DOI: 10.1007/BF00881714 · Zbl 0828.68109 [6] DOI: 10.2172/10129052 [7] 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/
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.