Moufang Unit Loops Torsion Over Their Centres

Abstract

Let L be an RA loop, that is, a loop whose loop ring in any characteristic is an alternative, but not associative, ring. We find necessary and sufficient conditions for the (Moufang) unit loop of RL to be torsion over its centre when R is the ring of rational integers or an arbitrary field. Over a field, torsion over the centre turns out to be equivalent to torsion of bounded exponent.