Von Staudt's theorem revisited (Q2350250)

The classical von Staudt theorem characterizes mappings of the projective line which preserve harmonicity as products of perspectivities. The author proves a version of this theorem for the projective line over a ring \(R\). Let \(M\) be a free \(R\)-module of rank 2. The points of the projective line \({\mathbb P}(M)\) are all free submodules of \(M\) of rank 1 which are direct summands of \(M\). Two points are called distant if their direct sum equals \(M\). The distant relation induces the structure of a simple graph on \({\mathbb P}(M)\), and the connected components of this graph are called the connected components of \({\mathbb P}(M)\). Assume that 2 is a unit in \(R\) and that for any five (not necessarily distinct) points \(p_1,\dots,p_5 \in {\mathbb P}(M)\) which are distant to some point \(p_0 \in {\mathbb P}(M)\), there exists \(p \in {\mathbb P}(M)\) which is distant to \(p_0,\dots,p_5\). Let \(M'\) be a free module of rank 2 over a ring \(R'\) and let \(\mu:{\mathbb P}(M) \to {\mathbb P}(M')\) be a mapping which preserves harmonic position of points. Then, the restriction of \(\mu\) to any connected component of \({\mathbb P}(M)\) is, after choosing suitable bases of \(M\) and \(M'\), induced by a Jordan homomorphism from \(R\) to \(R'\).