What Are R-Modules Used For?

Kind of a simple question, but what exactly are R-modules used for? Do they have any engineering applications?

EDIT:

If it helps, I'll give some more context to the question...

I am a graduate student researcher in computer architecture, a subfield of computer engineering. Specifically, I do research on the best way to build future general purpose processors (stuff like the Intel i7).

One thing I am looking into is if it is possible to apply mathematics to improve the design of CPUs. That is, can we use concepts from mathematics to improve the execution of general purpose programs on hardware. CPUs are a massive engineering design problem, and where exactly we could improve the design by applying math isn't entirely clear.

What I don't have is a very deep mathematical background. I have taken an introductory abstract algebra course and one in coding theory. I've also read a number of coding theory papers...

I know that other electrical engineering subfields like communications and compressed sensing have successfully applied elements of linear algebra and abstract algebra and have gotten very good results.

The fact that this particular question spans both engineering and mathematics makes it both hard to formulate and to discuss with people. I'd be happy to talk about it in more detail, but I'm not entirely sure what the best forum would be for that.

At least for now, I figured a good place to start would be to see if other people have successfully used some of the more abstract math concepts in engineering systems. One of the few I am aware of are R-modules, so I figured I'd ask if anyone knows of some engineering uses of them...

Solution 1:

Lest this stay unanswered I compose a list of the applications mentioned by various commenters. This is so obviously a CW-answer that everybody is welcome to add more examples. I am not making any claims about the relative importance of the list items.

1. Every vector space is a module, and you will have no trouble finding applications of vector spaces in a wide variety of fields.
2. The study of the set of solutions of systems of linear differential equations with constant coefficients is facilitated by the realization that they form an $\mathbf{R}[D]$-module.
3. In the theory of error-correcting codes, decoding algorithms for certain codes use Gröbner bases of modules over the ring of (univariate/bivariate) polynomials.
4. In telecommunications engineering, signal constellation design is facilitated by the use of modules over an algebraic number field.
5. In cryptography the construction of the NTRU cryptosystem similarly uses a structure that IIRC is best viewed as a module over the ring of modular polynomials.
6. Representation theory for groups uses module theory, and as a consequence everything that uses representation theory should be mentioned. For example, theoretical physics gets mileage out of this.