On classifying affine Barbilian spaces (Q1096883)

The author considers the representation of an affine Barbilian space by means of the corresponding affine geometry over a unitary free module, i.e. \(A_{FF}(M_ R,B)=(P,L,\phi,\|)\), where \(M_ R\) is a free unitary right R-module over an arbitrary ring R with identity, B is a Barbilian domain; \(P=set\) of points, \(L=set\) of lines, \(\phi =relation\) of non-neighbouredness, \(\| =parallelism\). In this way, he characterizes algebraic properties by geometric ones, and viceversa. The following are two of the results: a) \((P,L,\phi,\|)\) is linear unique (any two different points are contained in at most one line) iff R has no zero-divisors. - b) \((P,L,\phi,\|)\) is linear connected (any two different points are contained in at least one line) iff \(BR=M_ R.\) He also proves that the subclass of affine Hjelmslev spaces is characterized by the transivity of the neighboured relation, i.e. complementary to \(\phi\).