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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?

mma
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    You can do (most of) linear algebra over non-commutative division rings — you just need to be careful not to commute scalars. When you do, you call the objects you study vector spaces. – Mariano Suárez-Álvarez Aug 04 '22 at 06:43
  • What Mariano said. Over a non-commutative division ring you need to add left/right to many a definition for it to play out well. For example, if $\Bbb{H}$ is the ring of (Hamiltonian) quaternions, it is easy to exhibit two elements of $\Bbb{H}^2$ that are linearly independent as element of the left $\Bbb{H}$-module, but linearly dependent as elements of the right $\Bbb{H}$-module. It may be a bit confusing as to why the language of vector spaces (as opposed to modules) is often used. It may be because every finitely generated (left) module is free. – Jyrki Lahtonen Aug 04 '22 at 08:06

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