A necessary and sufficient condition so that two planar ternary rings induce isomorphic projective planes

摘要

It is well known that two skewfields coordinatize isomorphic projective planes if and only if they are isomorphic. In this paper, we study the relation between two planar ternary rings (PTRs) coordinatizing isomorphic projective planes in the general case. The fact that every projective plane can be coordinatized by an Intermediate Ternary Ring, as well as by a Planar Ternary Ring With Zero (PTRZ), or by a Hall Ternary Ring, (see [3], [4], [5]), has forced us to give a method of coordinatization where all the PTRs obtained by using these methods can be recovered. This method has lead us to introduce the notion of comparison for PTRs not necessarily with zero, which we have used with the notion of inversion of PTRZs (see [1]) to give a necessary and sufficient condition so that two PTRs coordinatize or induce isomorphic projective planes. The well known theorems on skewfields, division rings and quasifields can therefore be deduced from this condition as particular cases.