Hmm, I'll need to think about that. Thank you!

I suppose a better way of phrasing my point is that it would seem that much of the structure that arises in eigenvalue theory specifically is making use of the fact that all of the submodules of a vector space are free, so if this does not property hold for a given module then it makes it unlikely that one could find analogies to the nice structure of eigenvalues/eigenspaces in said module.

Actually, understanding that the much of the theory of eignenvales and eigenspaces assumes that one is working with a free modules with the feature that all of its submodules are also free gave me an incredibly helpful starting point, so thank you. :-)

Hmm, I'll need to chew on that for a while. Thanks a lot! :-)

But one could say the same thing about eigenvectors: they are just one-dimensional linear subspaces that happen to be invariant under the action of O and not more. And yet there has been a great body of theory with all sorts of interesting results with practical applications developed about them. Is there really no analogous body of theory for modules, or perhaps modules with some additional structure such as a dot product and norm?