Von Staudt’s theorem revisited

Abstract

We establish a version of von Staudt's theorem on mappings which preserve harmonic quadruples for projective lines over (not necessarily commutative) rings with "sufficiently many" units, in particular 2 has to be a unit. Mathematics Subject Classification (2010): 51A10 51C05 17C50Key words: harmonic quadruple, harmonicity preserver, projective line over a ring, Jordan homomorphism

Germán Ancochea's work on projectivities, harmonicity preservers and semi-homomorphisms

“…A proof of this result can be derived from B. V. Limaye and N. B. Limaye [110], despite the fact that their work from 1977 is mainly about commutative rings. A formal proof under slightly weaker assumptions was given by the author [63]. Already in 1971, N. B. Limaye [112], [113] proved a version for commutative local rings [105, pp. 280f.]…”

“…As a consequence, one may select arbitrarily one Jordan homomorphism R → R per component in order to create a harmonicity preserver P(M) → P(M ). The material from the last two paragraphs forms the foundation for the version of von Staudt's Theorem in [63,Thm. 1].…”

A generalisation of von Staudt’s theorem on cross-ratios

“…Any result in this spirit is now called a von Staudt theorem. Over the years this theorem was generalised by relaxing the assumptions on F, for instance for F a skew-field or a ring with some additional assumptions, see the introduction in [Hav15] for a survey of results in that direction.…”