## Radical parallelism on projective lines and non-linear models of affine spaces.(English)Zbl 1027.03009

Let $$R$$ be an (associative) ring with $$1\neq 0$$. A point $$p$$ is a submodule $$p= R(a,b)$$ of $$R^2$$ if $$(a,b)\in R^2$$ admits $$(c,d)\in R ^2$$ such that $$(*)$$ the pair $$(a,b)$$, $$(c,d)$$ is a basis for the free $$R$$-module $$R^2$$. The set of all points is called the projective line $${\mathbf P}(R)$$ over $$R$$. Points $$p$$ and $$q$$ are called distant if one has $$p= R(a,b)$$ and $$q=R(c,d)$$ such that $$(*)$$ holds true. These concepts were introduced by E. Salow [Math. Z. 134, 143-170 (1973; Zbl 0256.50002), page 145]. A point $$p$$ is called radically parallel to a point $$q$$ if each point $$x$$ that is distant to $$p$$ is also distant to $$q$$. Let $$\overline{R}$$ denote the factor ring of $$R$$ by its Jacobson radical. The canonical homomorphism induces a mapping of projective lines $${\mathbf P}(R) \rightarrow {\mathbf P}(\overline{R})$$. The authors prove that two points of $${\mathbf P}(R)$$ have the same image if and only if the points are radically parallel. When a field $$K$$ in the center of $$R$$ is given one obtains a chain geometry. The affine space given by the $$K$$-vector space $$R$$ is a sub-structure of that chain-geometry. The authors find automorphisms of the chain geometry yielding nonlinear models of the affine space, e.g. the parabola-model of the real affine plane.

### MSC:

 03B30 Foundations of classical theories (including reverse mathematics) 51C05 Ring geometry (Hjelmslev, Barbilian, etc.) 51B05 General theory of nonlinear incidence geometry 51N10 Affine analytic geometry

Zbl 0256.50002
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