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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

Citations:

Zbl 0256.50002
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