Units in alternative integral loop rings
Given the ring \(\mathbb{Z}\) of integers and a loop \(L\) the integral loop ring \(\mathbb{Z} L\) is the free \(\mathbb{Z}\)-module with basis \(L\) and multiplication defined distributively from the multiplication of \(L\). In a ring a unit is an element which is invertible. \(RA\) loops are loops whose loop rings over any associative ring of characteristic \(\neq 2\) are alternative but not associative. A subloop \(H\) of an \(RA\) loop \(L\) is torsion if all elements in \(H\) are of finite order. The authors study the ``isomorphism problem'': When the ring isomorphism \(\mathbb{Z} L\cong\mathbb{Z} M\) implies the loop isomorphism \(L\cong M\)? They prove that the integral loop ring of a finitely generated \(RA\) loop determines the loop, i.e. \(\mathbb{Z} L\cong\mathbb{Z} M\Rightarrow L\cong M\). Also they deduce some properties of the torsion units of the integral loop ring \(\mathbb{Z} L\) of a finitely generated \(RA\) loop \(L\).
scientific article; zbMATH DE number 1029203