Extending the Concept of Chain Geometry

TL;DR

This paper generalizes chain geometry by defining $(K,R)$ over rings with a subfield, revealing new conditions for chain determination and describing residue structures via affine spaces.

Contribution

It extends the concept of chain geometry beyond algebraic rings, introducing new criteria for chain uniqueness and residue analysis.

Findings

Chain is uniquely determined by three points iff $K$'s multiplicative group is normal in $R$'s units.

Chains through a fixed point form compatibility classes linked to affine space structures.

Residue at a point can be described using a family of affine spaces with shared points.

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.