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Intuition for modules other than "vector spaces but for rings" ?

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Think of them as an abelian group with an associated ring action that works "nicely" (read: follows the distributive property). For example, all abelian groups are natural Z-modules, since they are abelian groups with a well defined Z-action (for n in Z, let ng be g added to itself n times).

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It's better to think of vector spaces as upgraded modules, rather than modules as downgraded vector spaces. This is because vector spaces have a ton of structure that modules don't. For example, all vector spaces are "free" over the base field, whereas modules are not. This means that generators of a module don't neatly form a basis like in the case of vector spaces.  Modules can also have torsion (for example, Z/6Z is a Z-module where every non-prime number has torsion).

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For practical examples for why this may be useful, I'll broadly say that algebraic objects "acting" on other algebraic objects is in some sense the motivating principle behind all abstract algebra. Groups, for example, were (in part) created to formalize the notion of a group action on topological spaces. Similarly, "ring" actions are super useful in modern mathematics. For example, the continuous (or smooth) functions on a manifold form a ring. It may be helpful to see how this ring acts on the manifold itself, where addition of points is locally defined (since a manifold is locally Euclidean, where adding points is just adding vectors). This means that any open Euclidean patch U of a manifold M naturally has an associated ring action via the ring of smooth functions on U, so U is a module over this ring (more generally, the whole manifold has a "sheaf" of modules parameterizing these patches, and this naturally extends to all kinds of geometric objects).

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Another nice example of a module is actually a vector space, though not in the way you think. Let V be a finite dimensional vector space over a fixed field k. Let T be an endomorphism on V (think of T as a matrix). k\[x\] is the ring of polynomials in one variable with coefficients in k. Any polynomial f in k\[x\] can act on vectors of V via applying the fixed endomorphism. For example, x\^2 + 1 acts on a vector v by sending v to (T\^2(v) + v), and x\^3 + 2x\^2 acts on v by sending v  to T\^3(v) + 2T\^2(v). This is a linear action, so V has a k\[x\] module structure. This is useful because k\[x\] is a PID and V is finitely generated over k\[x\] so we can use the structure theorem to decompose V as a k\[x\] module. Doing this in invariant factor form yields the minimal polynomial of T, and more generally, defines the jordan canonical form of T as a matrix.
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If you know geometry, modules (over commutative rings) are like vector bundles — elements in a module are like sections of a vector bundle. You can think of the base ring as the ring of functions, and that naturally defines scalar multiplication on sections of a vector bundle.

Furthermore, you can base change a module to some local ring and its residue field which gives  you a vector space, and it’s precisely the fiber of the vector bundle at a point.

There are a lot of results about modules in commutative algebra that can be interpreted geometrically and they are central to the study of algebraic geometry.
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It's a bit like asking what is the point of studying commutative rings , when they are just fields where not all nonzero elements are invertible. The point is that there is a rich theory of commutative rings that is applicable to  specific objects of interest, e.g. polynomial rings in several variables over say the complex numbers and results involving irreducible homogeneous polynomials in these rings correspond to results about irreducible projective plane curves. More generally there is the Nullstellensatz which is a foundational result in the study of algebraic curves.

So my answer would be that this is a similar situation. There is a rich theory of modules and inside of it one can recover the usual facts about vector spaces from linear algebra, but also much more.

To give a concrete example of the utility of studying modules, there is the structure theorem for finitely generated modules over principal ideal domains, which is a useful classification theorem to know for lots of other mathematics, like homological algebra. You can also prove things like the Jordan Canonical Form theorem by applying the structure theorem to k[T] modules given by some k vector space V, where k is an algebraically closed field, you define these modules by fixing some linear transformation L:V->V, and define multiplication by T as T·v =L(v), extended in the natural way as P(T).v = P(L)(v). By applying the structure theorem to this module you obtain information about L.
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The concept of a module allows algebraists to put two different-looking objects on an equal footing: an *ideal* I in a commutative ring R (with identity, of course) and the *quotient ring* R/I can both be viewed as the same type of object: they are both R-modules.  This allows arguments involving ideals and quotient rings for R to be formulated in a more uniform way as arguments about R-modules.


In the setting of geometry, if M is a smooth manifold then the space of all (smooth) vector fields on M is a module over the ring C^(∞)(M) of smooth real-valued functions on M.  More generally, when M is connected the space of all sections of a smooth vector bundle on M (vector fields are sections of the tangent bundle) is precisely what the finitely generated C^(∞)(M)-modules of a certain type look like. Read about the Serre-Swan theorem on Wikipedia. Maybe that is beyond what you're already familiar with, but it is a good example of the fact that modules don't show up in math only where algebraists are working.
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Lattices.

If you just take all the integer points in the Cartesian plane, then you have a Z-module. You can do similar stuff that you would with vector spaces, such as add two of them together, but there are limitation. You can't scale by an arbitrary factor, you can only scale by an integer. Not every rotation is valid, only rotations by multiples of pi/2. Moreover, if you multiply the lattice by 2, you don't hit every lattice point as some, such as (3,1), are missed.

This last point is important because of another important feature: Torsion.

Let's say that M is the integer grid above, then 2M is a sub-lattice of M - the set of all integer points on the plane with even components. You can then look at the action of 2M as, in a sense, living OVER M creating the quotient module M/2M. This is effectively the module (0,0), (1,0), (0,1), (1,1). This has torsion because any element in this module times 2 is zero. But you can look at it as one grid inside another grid where you're guaranteed to hit the smaller grid when you add any two points together. Another thing to note is that you can turn this into a Z/2Z module, viewing it as a Z/2Z-grid really easily because the actin of 2Z on M generates 2M (or, 2Z is the annihilator of the quotient group).

More complicated rings lead to more complicated structure, but most of the time the rings are nice enough to get nice enough results for what you're doing.
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Group representations give you modules over the group ring.

The set of sections in a vector bundle over a manifold is a module over the ring of smooth functions.

When studying Lie algebras you can think of representations as modules over the universal enveloping algebra.
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Currently, you're starting by thinking of vector spaces as somehow more intuitive or fundamental than modules.

Quite a lot of objects are modules. They're basically groups (that too abelian) with a scalar multiplication, so you can see how simple their structure is.

As you know vector spaces are modules. If you know about ideals in a ring, they are also modules. A module with a multiplication structure gives an algebra.

Modules, in some sense, are the most basic notion of what a linear combination of generators means. Imagine associating coefficients to the generators a, b, c as a tuple (x, y, z). Basically, the output is xa + yb + zc. Now, in order for this construction to work, you need tuples like (x, y, z) to be able to add to each other (imagine adding xa + 0 + 0 and x'a + 0 + 0) and being able to multiply (because scalar multiplication). This sort of forces a ring structure on your coefficients. The abelian group structure of the generators makes it a simple case when studying commutative objects.

It's the same concept as "vector spaces but for rings" but I gave a little elaboration on a way to look at vector spaces.
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There's a lot to unpack in the "vector space but for rings". You can begin by pondering why are we interested in giving a vector space structure to groups or rings? Vector spaces input the concept of linearity and orthogonality in the picture and thus allow us to do some analysis (representation theory).
by
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Might be worth pointing out, reducing the axioms of a space isn't a "downgrade" because now those axioms apply to many more objects than the original definition. In this case, Vector spaces are a small subset of all modules, and they're all free modules (i.e. all vector spaces have a basis with axioms of choice). When you look at all modules, there are modules which are finitely generated but not free, torsion modules, projective and injective modules, etc., basically a lot of extra interesting structure that you just don't get when every nonzero scalar is invertible.

Also if you study rings, the modules over a ring have a lot of connection with the properties of the ring. Modules over a PID are a good place to start.
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Quasi-coherent sheaves over Spec(R), of course.

So, "like vector bundles, but you actually get an Abelian category".

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