Coordinatization of generalized affine spaces (Q1378895)

A generalized affine space is defined as a quadruple \({\mathcal A} =(P,G,//, \%)\) with \(P\) the point-set, \(G\) the line-set, \(//\) a parallelism and \(\%\) a distant relation, satisfying a list of axioms. A characterization is given for those generalized affine spaces of dimension \(n(n\geq 3)\) which are naturally induced by a module over a ring. Moreover it is proved that the ring is a domain if and only if any two distinct points are distant and the space satisfies a version of the little Desargues' axiom.