TITLE: Generalization of eigenvalues/vectors to modules?
QUESTION [3 upvotes]: What is the generalization of eigenvalues/vectors to modules?
To be specific, given a "vector" v in a module over some ring, and a linear "operator" O from the module to itself (please feel free to correct my terminology :-) ), I would like to learn what we know about problems of the form
O v = k v
where k is a member of the same ring.
I have been looking through a lot of books and online resources about modules, but I am having trouble finding the answer to this question, and I am guessing that it is probably because I don't know what the name of the thing is that I should be looking for.
Edit:  Fixed a typo -- thanks Boris!  (I said that O was a map from the ring to itself when I meant it was a map from the module to itself.)
Update:  To be clear, I would also be happy with an answer of the form: there is not a good generalization of eigenevalues for modules with no additional structure at all, but there is if you can assume the additional structure X, where X is, say, a dot product, a norm, an involution operator, etc.

REPLY [8 votes]: There is another way to look at this. Let $K$ be a commutative ring and $M$ a $K$-module.
Then giving a $K$-linear endomorphism of $M$ is equivalent to an action of the polynomial ring $K[x]$ on $M$. Then the question becomes: what is the structure of $M$ as a $K[T]$-module? The classification of finitely generated $K[x]$-modules in the case $K$ is a field is a well-known result that used to be taught to undergraduates.
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