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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...

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Anyway, this is a very broad question. It would help a lot for us if you gave us some context (your abstract algebra background, why you want to know about modules) to help narrow down your question. –  Qiaochu Yuan Apr 23 '12 at 3:49
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@QiaochuYuan: Well, I'm not entirely sure as I'm a bit fuzzy on what they are...But I suppose what I mean when I say engineering applications is are R-modules used in systems such that they solve a real world problem? For example, finite fields are used to construct error correcting codes which allow us to transmit data over lossy lines and then reconstruct the resulting data with less overhead and better reliability than sending the message multiple times until it is received. And I guess finite fields have some applications in encryption. Are R-modules used for any applications like those? –  Adam Apr 23 '12 at 3:53
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$R$-modules are useful because they generalize vector spaces. Any use you know of the theory of vector spaces (e.g. all of linear algebra), is a use of a particularly nice and well-understood part of the theory $R$-modules. Much of the motivation of $R$-modules comes from wanting to apply the ideas of linear algebra in settings where nonzero "scalars" might not be invertible (so one cannot just use linear algebra). I cannot think of any engineering applications of the theory of $R$-modules beyond linear algebra. I would not advise general study of the theory of $R$-modules. –  leslie townes Apr 23 '12 at 5:31
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Since you seem to be interested in telecommunications applications I will offer space-time coding (or the design of signal constellations for multiantenna radio transmission) as a topic, where theory of modules has played a role. The earliest constructions did not really need the language of modules. Later richer codes were found by intelligent search. But not much could be proven about them until it was realized that they were modules over a larger than expected ring. That opened up new ways of looking at it... –  Jyrki Lahtonen Apr 23 '12 at 5:39
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... and then it became easier to find larger constructions inaccessible by search methods. This is IMHO typical of applications of abstract algebra: they bring order to chaos. The order brought about by algebraic structures makes the analysis of the system simpler. And it enables some further developments. I agree with Qiaochu that DEs form a nicer application. There we also have the same phenomenon: the technique of solving DEs can be learned without knowing a thing about $\mathbf{R}[D]$-modules, but if you want the most concise explanation... –  Jyrki Lahtonen Apr 23 '12 at 5:49

1 Answer 1

up vote 2 down vote accepted

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. The study of the set of solutions of systems of linear differential equations with constant coefficients is facilitated by the realization the they form a $\mathbf{R}[D]$-module.
  2. In the theory of error-correcting codes, decoding algorithms for certain codes use Gröbner bases of modules over the ring of (univariate/bivariate) polynomials.
  3. In telecommunications engineering, signal constellation design is facilitated by the use of modules over an algebraic number field.
  4. 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.
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