Alternative loop rings with solvable unit loops (Q5938598)

Let \(L\) be a loop, \(R\) a commutative, associative ring of characteristic \(\neq 2\), \(RL\) the loop ring. A loop \(L\) is called an RA loop if \(RL\) is an alternative ring. If \(L\) is an RA loop, the set \({\mathcal U}(RL)\) of units in \(RL\) is a Moufang loop (containing \(L\)).NEWLINENEWLINENEWLINEIn this paper the authors compare the properties of \(L\) and \({\mathcal U}(RL)\) and give necessary and sufficient conditions for the solvability of \({\mathcal U}(RL)\) when \(R=\mathbb{Z}\) is the ring of rational integers or a field of arbitrary characteristic.