From a linear algebraic viewpoint, projective space is the set of the 1-dimensional subspaces of a vector space. Specifically, if the vector space has dimension 3 then its projective space is called a projective plane.
According to Wikipedia, Desarguesian projective planes can be constructed from a three-dimensional vector space over a skewfield. The extension of this definition to greater dimensions is that Desarguesian projective spaces can be constructed from a vector space over a skewfield. Skewfield is the other name of division ring and division rings are the same as fields except commutativity of multiplication.
But such thing as vector space over a skewfield doesn't exists. If the field in the definition of vector space is replaced by a ring then its name is no longer "vector space" but "module". Is it true that a Desarguesian projective space is the set of 1-dimensional submodules of a module over $K$ where $K$ is a division ring?
So, I think, Desarguesian projective space is the extension of the notion of projective space in the same way as the extension of vector spaces to modulules over division rings.
Is my understanding correct?
EDIT:
If yes, then every projective space is Desarguesian. But then what are non-Desarguesian projective spaces?