Symmetric Units in Alternative Loop Rings

Abstract

Let L be an RA loop, that is, a loop whose loop ring in any characteristic is an alternative, but not associative, ring. For α = ∑ αℓℓ in a loop ring RL, define α♯= ∑ αℓℓ-1and call α symmetric if α♯= α. We find necessary and sufficient conditions under which the symmetric units are closed under multiplication (and hence form a subloop of the loop of units in RL) when R has characteristic two and when R= Z, the ring of rational integers.