Regular homomorphisms of generalized projective planes (Q580705)

The author defines a (generalized) projective plane as an incidence structure \(\Pi =({\mathcal P},{\mathcal L},I)\) equipped with a relation ``distant'' on the set of points \({\mathcal P}\) and on the set of lines \({\mathcal L}\) so that the following axioms and their duals hold: (PO) If A distant B then A, B have a unique common line (A,B). (P1) If \(hIA,\quad B,\) \(gIB,\) A distant B and g distant h, then A distant B' and g distant h' for each point B'Ig and each line h'IA. (P2) There is at least one point on every line. If AIc then BIc and B distant A for some B. (P3) If A distant B, then CI(A,B) and C distant A, B for some C. Every commutative ring R with 1 supplies a model \(\Pi\) (R). A homomorphism \(\phi\) of \(\Pi\) is called regular if A distant B and \(c\phi\) IA\(\phi\), \(B\phi\) imply \(c\phi =(A,B)\phi\) and the dual holds. One of the main results is the following: Suppose \(\Pi\) (R) satisfies condition (U): Given A, b, there is a point B and a line a so that BIb, B distant A, aIA and a distant b; then every surjective regular homomorphism \(\phi\) of \(\Pi\) (R) is induced by an ideal of the ring R; the image of \(\Pi\) (R) under \(\phi\) is again a projective plane, and \(\phi\) preserves the relation ``distant''.