I know that it is possible to build a projective plane over any field. I also know that it is possible to build a projective plane over Hamilton's quaternions.
My question is: Is it possible to build a projective plane over a ring which is not a division ring?
In general: what is the minimal algebraic structure that allows us to build a projective plane?
Very good question!
You can make a projective plane out of any division ring, and it turns out that the projective planes that are coordinatizable by division rings (the ones you can build out of division rings) are exactly the Desarguesian projective planes.
The coordinatization of projective planes more general than that also leads to interesting results. The one to mention, I think, is that an algebraic structure called a planar ternary ring can be used to coordinatize projective planes. The nicer the plane is, the closer it will come to being a division ring.
An especially useful set of notes I found while educating myself on this latter part are notes by Timothy Vis:
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