Affine Barbilian-Ebenen II

Abstract

In part I we proved that every affine Barbilian plane is up to isomorphisms an affine geometry over a Z-ring R as defined in the introducing abstract there. Now we carry out that conversely all affine ring geometries over a Z-ring are affine Barbilian planes and represent their automorphisms algebraically as semilinear bijections. Finally, we present several classification theorems as, for instance, that the class of Desarguesian affine planes coinzides with the class of affine Barbilian planes, satisfying the additional axiom that two different points are always non-neighboured. The weaker condition that there is always exactly one line passing through two different points corresponds with the fact that the underlying ring is a right Bezoutring. There is at most one line passing through two different points iff the corresponding ring R has no zero-divisors.