NORMALITY OF f-UNITARY UNITS IN AN ALTERNATIVE LOOP RING

Abstract

Let L be an RA loop, that is, a loop whose loop ring in any characteristic is an alternative, but not associative, ring. Let f : L → {±1} be a homomorphism and for α = ∑αℓℓ in the integral loop ring Z L, define αf = ∑αℓf(ℓ)ℓ-1. A unit u ∈ Z L is said to be f-unitary if uf = ±u-1. The set [Formula: see text] of all f-unitary units is a subloop of [Formula: see text], the loop of all units in Z L. In this paper, we find necessary and sufficient conditions for [Formula: see text] to be normal in [Formula: see text].