Extending the Concept of Chain Geometry

Abstract

We introduce the chain geometry $\Sigma (K,R)$ over a ring $R$ with a distinguished subfield $K$, thus extending the usual concept where $R$ has to be an algebra over $K$. A chain is uniquely determined by three of its points, if, and only if, the multiplicative group of $K$ is normal in the group of units of $R$. This condition is not equivalent to $R$ being a $K$-algebra. The chains through a fixed point fall into compatibility classes which allow to describe the residue at a point in terms of a family of affine spaces with a common set of points.

Cited by in corpus (9)

• Projective Representations I. Projective lines over rings
• The Connected Components of the Projective Line over a Ring
• Jordan homomorphisms and harmonic mappings
• The Dual of a Chain Geometry
• Projective Representations II. Generalized chain geometries
• Affine Spaces within Projective Spaces
• Quadrangular sets in projective line and in Moebius space, and geometric interpretation of the non-commutative discrete Schwarzian Kadomtsev-Petviashvili equation
• Divisible designs from twisted dual numbers
• Linear sets in the projective line over the endomorphism ring of a finite field