Let $L$ be an RA loop, that is, a loop whose loop ring in any characteristic is an alternative, but not associative, ring. Let $f:L\to\{\pm1\}$ be a homomorphism and, for $\alpha=\sum\alpha_\ell\ell\in\mathbb{Z}L$, define $\alpha^f=\sum f(\ell)\alpha_\ell\ell^{-1}$. Call $\alpha$ f-unitary if $\alpha^f=\alpha^{-1}$ or $\alpha^f=-\alpha^{-1}$. In this paper, we identify the RA loops $L$ with the property that all units in $\mathbb{Z}L$ are $f$-unitary. Along the way, we extend a famous theorem of Higman to a case still undecided in group rings.
AMS 2000 Mathematics subject classification: Primary 20N05. Secondary 17D05; 16S34; 16U60
