Generalization of eigenvalues/vectors to modules? 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.
 A: 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.
A: In the case of commutative rings, you can view spectra as points in the quotient of $End_R(M)$ by the conjugation action of $Aut_R(M)$.  You can use the tensor product to turn this into a quotient of a scheme by a group.  If $M$ is locally free of rank $n$, the coefficients of the characteristic polynomial (say, viewed as traces of $\wedge^i O$ on $\wedge^i M$) give you a map to affine $n$-space over the spectrum of $R$.  You can think of this as a space of elementary symmetric polynomials in eigenvalues.  If you take your operator $O \in End_R(M)$, and send it to a point in this space, I suppose an eigenvalue is what you get by lifting to an element in the $S_n$-orbit in the affine space of roots, and projecting to a coordinate.  These don't exist globally.
This sort of construction arises when studying the Hitchin map.
(Minor comment: Darij's claim that the eigenvalue map is discontinuous uses an implicit assumption that the set with three elements should be endowed with the discrete topology.)
A: This just means that the submodule generated by $v$, i.e. $\lbrace rv\mid r\in R\rbrace$, is invariant with respect to the operator $O$ (which acts rather in the module than the ring $R$), no more.
