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.


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
Full Text: arXiv EuDML