The authors identify the RA (ring alternative) loops \(L\) with the property that all units in \(\mathbb{Z} L\) are \(f\)-unitary. (Call \(\alpha\) \(f\)-unitary if \(\alpha^f=\alpha^{-1}\) or \(\alpha^f=-\alpha^{-1}\).) Along the way, the authors extend a famous theorem of Higman to a case still undecided in group rings.
The best reference for information about RA loops and their loop rings is the monograph by E. G. Goodaire, E. Jespers and C. Polcino Milies, Alternative loop rings [North-Holland Math. Stud. 184 (1996; Zbl 0878.17029)].