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


17D05 Alternative rings
20N05 Loops, quasigroups