A generalization of the cross-ratio over a ring (Q756123)

Let D be a (not necessarily commutative) field of characteristic \(\neq 2\), and \(P_ 1(D)\) the projective line over D. A well-known theorem of von Staudt and Hua says that the group of automorphisms of the geometrical structure defined on \(P_ 1(D)\) by its harmonic quadruples is isomorphic to the group of automorphisms and anti-automorphisms of D. Attempts to extend this result to the more general case of an (associative) ring can be found in articles by \textit{C. Bartolone} and \textit{F. DiFranco}, [Math. Z. 169, 23-29 (1979; Zbl 0413.51006)] and \textit{N. B. Limaye} [Math. Z. 121, 175-180 (1971; Zbl 0215.501)] for commutative rings, and by \textit{C. Bartolone} and \textit{F. Bartolozzi} [Rings and Geometry, NATO ASI Ser., Ser. C 160, 353-389 (1985; Zbl 0612.51007)] and \textit{N. B. Limaye} and \textit{B. V. Limaye} [Arch. Math. 27, 102-109 (1977; Zbl 0351.50007)] for the non-commutative case. In the present paper the authors generalize the result to the case of the projective line over a (not necessarily commutative) ring.