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I'm looking for a non-trivial1 example of a module that would be recognizable to a non-mathematician. I.e. I'm looking for examples of modules that one may come across in "the real world".

The closest I can come up with are vector spaces, but I'd prefer an example that is not a vector space (i.e. one where the module's ring is not in fact a field).

(If this question were about groups instead of modules, a good answer would be the group of rotations of a cube, or the group of permutations of three objects. Granted, most non-mathematicians would not construe such rotations and permutations "group-theoretically", but the ideas would be "recognizable" to them, at least in the sense I have in mind. In fact, popular accounts of group theory often resort to such groups as the first examples to present to non-mathematical readers.)

1 Here I'm using non-trivial in an ad hoc, non-standard sense. By "non-trivial module" I don't mean one with more than one element, but rather one that is not merely a ring. (E.g., the ring of integers $\mathbb{Z}$ can be regarded also as a module, with $\mathbb{Z}$ playing both the role of the ring and the Abelian group that go into the definition of a module. This "module" $\mathbb{Z}$ would certainly be familiar to non-mathematicians, but is among the ones that I'm trying to exclude with the qualifier "non-trivial".) If there's a better name for such "non-trivial" modules, please let me know.

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Are more general abelian groups permissible? I'm thinking predominantly of lattices $\mathbb{Z}^n$, but perhaps that isn't sufficiently less trivial than $\mathbb{Z}$ for your purposes. –  Matt Pressland Jul 4 '13 at 14:26
@MattPressland: I like your suggestion; please post it so that I can accept it. –  kjo Jul 4 '13 at 14:52

4 Answers 4

up vote 1 down vote accepted

I think lattices $\mathbb{Z}^n$ (particularly for $n=2$ and $n=3$) are at least easy to visualize as examples. It's not immediately clear to me why non-mathematicians might have thought about them already, but at least the concept of moving around space in discrete jumps is a fairly natural one.

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If someone has five quarters, they can be considered as $\mathbb{F}_2S_5$- modules, acting by permutation and flipping (here, we assume the quarters all start as tails, and that adding two module elements corresponds to taking all tails, flipping as many as you need to get the first element, and then flipping as many as you would have needed to get the second element, giving you a third element. So adding any element to itself repeats the exact same sequence of flips twice, so returns to tails).

Group ring modules are everywhere. Molecules that have a non-trivial symmetry group have equations of motion/oscillation that are invariant under group rings where the group is the symmetry group.

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If $V$ is a vector space over $F$ and $T:V\to V$ is a linear operator, then $V$ is a module over the polynomial ring $F[x]$ (f(x)v=f(T)v). Moreover, this is true for abelian groups instead of spaces and $\mathbb Z$ instead of $F$.

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Have you read the question or at least the title? –  Martin Brandenburg Jul 5 '13 at 14:10
@ Martin Brandenburg: Yes, I read –  Boris Novikov Jul 5 '13 at 14:25

What about polynomial rings? They're modules over the ring of the coefficients.

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Have you read the question or at least the title? –  Martin Brandenburg Jul 5 '13 at 14:11

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