**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.

Read more about this topic: Intersection Theorem

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