Intersection theorem
In projective geometry, an intersection theorem or incidence theorem is an incidence structure consisting of points, lines, and possibly higher-dimensional objects and their incidences, together with a pair of nonincident objects A and B (for instance, a point and a line). The "theorem" states that, whenever a set of objects satisfies the incidences (i.e. can be matched up with the objects of the incidence structure in a way that preserves incidence), then objects determined by A and B must also be incident. An intersection theorem is not necessarily true in all projective geometries; it is rather a property which some geometries satisfy but not others.
For example, Desargues' theorem can be stated using the following incidence structure:
- Points:
- Lines:
- Incidences (in addition to obvious ones such as
: 
The implication is then 
—that point R is incident with line PQ.
Famous examples
Desargues' theorem holds in a projective plane P if and only if P is the projective plane over some division ring D—
. The projective plane is then called desarguesian.
A theorem of Amitsur and Bergman states that, in the context of desarguesian projective planes, for every intersection theorem there is a rational identity such that the plane P satisfies the intersection theorem if and only if the division ring D satisfies the rational identity.
- Pappus's hexagon theorem holds in a desarguesian projective plane
if and only if D is a field; it corresponds to the identity 
. - Fano's theorem (which states a certain intersection does not happen) holds in if and only if D has characteristic
; it corresponds to the identity a+a=0.
References
- L. H. Rowen; Polynomial Identities in Ring Theory. Academic Press: New York, 1980.
- S. A. Amitsur; "Rational Identities and Applications to Algebra and Geometry", Journal of Algebra 3 no. 3 (1966), 304–359.