Uniqueness of barbilian domains

Abstract

W. Leissner has developed a plane geometry over any Z-ring R, in which a point is an element of R×R and a line is a set of the form {(x+ ra, y + rb):r ∈ R} where (x,y) ∈ R×R and (a,b) is from a “Barbilian domain”, i.e., a set of unimodular pairs from R×R satisfying certain axioms. In this note we generalize results of W. Benz guaranteeing the uniqueness of Barbilian domains over several classes of commutative rings. The author wishes to thank Gordon Keller and Douglas Costa for fruitful discussions, the referee for his improvements, and the University of Virginia for its hospitality while this work was done.