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Projectivities over rings with many units. (English) Zbl 0466.51018
In this paper the author proves a generalisation of the fundamental theorem of projective geometry for the projective line over primitive rings (A ring is said to be “primitive” if for any polynomial \(f\) in \(R[X]\) whose coefficients generate \(R\) as an ideal, there exists \(\alpha\in R\) with \(f(\alpha) = \)a unit). Von Staudt’s theorem was proved for local rings by N. B. Limaye [Math. Z. 121, 175–180 (1971; Zbl 0215.50102)] and the proof in this paper is analogous to the proof by N. Limaye. Von Staudt’s theorem has also been proved by M. Kulkarni [Indian J. Pure Appl. Math. 11, 1561–1565 (1980; Zbl 0512.51001)] and by C. Bartalone and F. Di Franco [Math. Z. 169, 23–29 (1979; Zbl 0413.51006)]. By weakening the definition of harmonic quadruple they have obtained the same theorem for an arbitrary commutative ring. However, they do not prove Hua’s theorem which is an intermediate result proved in this paper.

MSC:
51N15 Projective analytic geometry
14A05 Relevant commutative algebra
14N05 Projective techniques in algebraic geometry
51M99 Real and complex geometry
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References:
[1] Artin E., Geometric Algebra (1957)
[2] Camillo V.P., Morita equivalence and the Fundamental Theorem of projective geometry
[3] DOI: 10.24033/asens.1334 · Zbl 0393.18012
[4] DOI: 10.1007/BF01113485 · Zbl 0215.50102
[5] DOI: 10.1007/BF01223897 · Zbl 0351.50007
[6] DOI: 10.1080/00927878008822495 · Zbl 0436.20031
[7] DOI: 10.1080/00927878108822573 · Zbl 0453.20041
[8] DOI: 10.1007/BF02564531 · Zbl 0176.17502
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