Isomorphisms of integral alternative loop rings
Let us call a - not necessarily associative - ring A left or right alternative iff \(x^ 2y=x(xy)\) or \((xy)y=xy^ 2\) for all \(x,y\in A\), resp., and alternative iff it is left and right alternative. - For an associative ring R with unity and a loop L 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 investigations under review are concerned with sufficient conditions on L such that the right alternativity of RL implies the alternativity of RL. By handling some nontrivial term transformations the authors succeed in proving the following results: for finite L and char \(R\neq 2\), if RL is right alternative it is alternative; if L is a Moufang loop the following statements are equivalent: RL is left alternative, RL is right alternative, RL is alternative. - The authors conjecture more generally that right alternativity of a loop ring always implies its alternativity.
scientific article; zbMATH DE number 4140001