I'm studying projective geometry in a basic course of geometry. My question is:
Is there an equivalent definition of projective space not of a vector space but of a module?
I think the basic definition is the same but the equivalence relation:
Let $A$ be a ring and let $M$ be a module over $A$. Let $v,w \in M$ be. We say $v \sim w$ if $\exists a \in A^{*}$ (unit element) such that $v = aw$.
I think the condition $a \in A^{*}$ is necessary for the simmetry of equivalence condition.
Is this construction studied in any topic of mathematics? If so, are there any books where to find it?
Thanks.
It is true that this is an equivalence relation. You could certainly study the properties of this equivalence relation, and some of them might be interesting. However, you certainly wouldn't get a projective space, and probably wouldn't find very much "geometry." It is really more of an algebraic question.
The nice thing about defining a projective space in terms of the subspaces of a vector space is that this definition satisfies the geometric axioms of a projective space: that every two point are contained in a unique line, and every two coplanar lines intersect in a unique point, as well as a non-triviality condition (https://en.wikipedia.org/wiki/Projective_space). I don't think this equivalence relation would give rise to any such geometric structure. Your "lines" would not have the regularities that they do over a field.
That said, the equivalence relation you describe is studied! If you define that same equivalence relation on pairs, it gives rise to "projective lines over a ring." The subject has a wikipedia article: https://en.wikipedia.org/wiki/Projective_line_over_a_ring
Another generalization that does exist is that of projective spaces on modules over division rings, rather than fields. In division rings, the multiplication does not need to commute, but division rings still give rise to projective spaces. The ever helpful wikipedia article on the projective plane describes this construction - it's basically the same as the construction of the projective plane over a field.
I don't know of any books on general projective lines over rings. The wikipedia article cites the paper: Blunck & H Havlicek (2000) "Projective representations: projective lines over rings". I might recommend books on projective geometry: one I know of is Robin Hartshorne's Foundations of Projective Geometry.