## Is a right alternative loop ring alternative?(English)Zbl 0696.17016

This article is one of a series of papers by E. G. Goodaire - mostly together with O. Chein and M. M. Parameter - which deal with generalizations of group rings to loop rings. Here the question is treated of whether a loop is - up to isomorphism - determined by its integral loop ring.
For a loop L and an associative ring R with unity the loop ring of L over (the coefficient ring) R is constructed precisely in the same way as a group ring and denoted by RL. The authors call a loop whose loop rings in characteristic $$\neq 2$$ are alternative but not associative RA loop (such a loop is necessarily a Moufang loop) and prove: for RA loops L and M where L is a torsion loop, i.e. the exact analogue to a torsion group, the isomorphy of $${\mathbb{Z}}L$$ and $${\mathbb{Z}}M$$ implies the isomorphy of L and M.
Let N be a normal subloop of a loop L and $$\phi$$ the ring homomorphism $${\mathbb{Z}}L\to {\mathbb{Z}}(L | N)$$ obtained by extending the canonical map $$L\to L | N$$; $$\phi$$ is an augmentation iff $$N=L$$. An automorphism $$\alpha$$ of $${\mathbb{Z}}L$$ is called normalized iff $$\alpha$$ preserves augmentation. - Let A be an alternative ring and U(A) the set of units of A. $$L_ a\rightleftharpoons x\mapsto ax$$ and $$R_ a\rightleftharpoons x\mapsto xa$$ are the left and right translation on A by a ($$\in A)$$, respectively. An automorphism of A is called inner it is contained in the group generated by $$\{L_ u,R_ u|$$ $$u\in U(A)\}$$. - By a nontrivial proof the authors reach the following interesting result: Let L be a torsion RA loop; to every normalized automorphism $$\zeta$$ of $${\mathbb{Z}}L$$ there is an inner automorphism $$\psi$$ of $${\mathbb{Q}}L$$ and an automorphism $$\sigma$$ of L with $$\zeta =\psi \circ \sigma$$.
Reviewer: W.Lex

### MSC:

 17D05 Alternative rings 20N05 Loops, quasigroups