Why this question? I try to understand connection between Module & Field more deeply to try to build a particular class of 'fixed points' of non-trivial affine space to 'extract' an affine subspace isometry.
This connection that I try to figure out is hard but I accept and propose a challenge
I start from this definition from Wiki
module over a ring is a generalization of the notion of vector space over a field
So I find that a 'connection' between modules and fields like algebraic structures should be somthing inside definition of Group Action $G$: Spaces of coinvariants or maybe an affine connection
also from wiki:
The group action is transitive if and only if it has exactly one orbit, i.e., if there exists x in X with G⋅x = X. This is the case if and only if G⋅x = X for all x in X.
The set of all orbits of X under the action of G is written as X/G (or, less frequently: G\X), and is called the quotient of the action. In geometric situations it may be called the orbit space, while in algebraic situations it may be called the space of coinvariants, and written XG, by contrast with the invariants (fixed points), denoted XG: the coinvariants are a quotient while the invariants are a subset.
I return now on field $\mathbb{F}$ definition
a field is a set, along with two operations defined on that set
But this definition for me is not useful, it very generical it tells me everything and at the same time nothing
I try to figure out the 'behavior' (behavior = two operations on set) of field $\mathbb{F}$ to represent same structure in a more open view concept: field $\mathbb{F}$ in terms of hyperplane, isometry plane and Reflection Relation between coordinate Space & vector Space since the field is the underlying structure to build coordinate space into vector space.
Normally people learn about a field acting on an abelian group first, and they call it a vector space.
If they are lucky, later they learn about rings, and define modules as abelian groups acted upon by rings, and they realize "oh, when the ring is a field, that is just a vector space."
They may also have the opportunity to study an action of a set $X$ on an abelian group (as seen here) and that is even more general.
Now, a lot of your post I can't make heads or tails of, but your last sentence makes sense to me, and if that is the most important thing you are asking about I can speak to it:
From this it seems to me you are asking how "field" relates to "geometry," and that, at least, has an interesting answer.
It turns out that given any division ring (=a possibly noncommutative field) $D$, $D\times D$ is a Desarguesian plane coordinatized by $D$, satisfying all the usual synthetic axioms of geometry.
The interesting bit is that given any Desarguesian plane (characterized by the synthetic axioms only) you can build a division ring that coordinatizes it. So division rings are precisely the rings that coordinatize Desarguesian planes.
There is a little bit of heuristic sense one can come up with that justifies why division rings are good rings to coordinatize geometries with. For one thing, the fact that every nonzero element is invertible allows you to take any nonzero element in a line through the origin and rescale it to match another nonzero element. Said another way the multiplicative group of $D$ acts transitively on lines. For another thing, the axioms for linear subspaces describe pencils of parallel lines (each one has a member going through the origin, and the others are parallels to it.) The abelian group operation allows for translations in the plane.
I think there may be one or two other things to say along those lines, and I'll add them as I recall them, if you turn out to be interested in this discussion.