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.

Whatexactly do you want to generalize? How will you tell if a proposed generalization isgood? What do you want to do with it? – Mariano Suárez-Alvarez♦ Nov 24 '11 at 6:17commutativerings. I suppose you want your operator to be at least $R$-linear, which means that is commutes with all scalar multiplications by elements of $R$. However if $k$ is not in the center of $R$, then multiplication by $k$ does not have this property, so there is no hope of finding any solutions for such $k$. – Marc van Leeuwen Nov 24 '11 at 13:36