Are there analogues of eigenvalues/eigenvectors for a ring homomorphism/endomorphism? My question is very simple. To put it in a context, a linear transformation is nothing but a homomorphism from a vector space to another. I usually visualize the action of a linear transformation by thinking of what it does to the unit hypersphere. Eigenvectors together with their eigenvalues describe the axes of action in this case.
I couldn't find an explicit reference to any sort of analogue of this phenomenon to ring homomorphisms/endomorphisms. Specifically, is there a standard way to describe the action of a ring homomorphism/endomorphism in the described sense? Even more specifically, what would be the analogue of an orthogonal transformation (with all eigenvalues modulus 1) in the realm of ring homomorphisms/endomorphisms?
edit: I believe now that I asked my question in a rather haphazard way. But I am keeping the first two paragraphs intact. I am not a mathematician and in the past few years I taught myself what it looks like almost completely ignoring technical details and first concentrating on the concepts. What I would be asking is in fact what the set of all ring homomorphisms/endomorphisms (from a ring to itself/to another one) look like? Can I introduce any metric/criteria to distinguish them, saying these are well behaving and these are not, just like I can do for linear transformations?
 A: You can't talk about the eigenvectors of a linear transformation between two different vector spaces. You can only talk about the eigenvectors of a linear transformation from a vector space to itself. So the analogous question is about ring endomorphisms, but ring endomorphisms are complicated in general and don't admit a description anywhere near as simple as the description of linear transformations. 
For example, understanding the behavior of ring endomorphisms of $\mathbb{C}(x)$ is highly nontrivial and leads to complicated phenomena like Julia sets. The elements of Galois groups are also ring endomorphisms, and understanding Galois groups is also a major problem (e.g. understanding the absolute Galois group of the rationals is a major goal of modern number theory). 
A: Look at the equation $Tx=\lambda x$. Since the $x$ is in both sides, it looks like $T:R\to R$ for the equation to make sense, and moreover we need to figure out what $\lambda$ is, whether it is just from $R$ or from some module action on elements of $R$.
I'll call $x$ and "eigenelement" of $R$. As with vector spaces, you'll have no problem verifying that eigenelements corresponding to the same eigenvalue $\lambda$ form a subgroup of $R$. In order to form a right ideal of $R$, though, we would need $T(xr)=\lambda xr$ for arbitrary $r\in R$. This is even unlikely when $R$ is commutative, for we have that $T(xr)=T(r)\lambda x$, and there is no guarantee that $T(r)=r$. So, hope is lost for "eigenideals" of $R$.
I suppose this means that you can talk about "eigensubgroups of $R$", but I've never heard anything like this leading to a useful decomposition of $R$. Certainly it would be unusual for it to turn out to play well with the multiplicative structure. 
Things get a little better if you allow yourself to consider a $K$ algebra $R$ for a commutative ring $K$. In this case, you would also be requring the ring endomorphism $T$ to be $K$ linear. Accordingly, $R$ is just a $K$ module, and your ring endomorphism is a fortiori a $K$ module homomorphism. That would allow $K$-eigenvectors to be described for $T$, but they would only be giving information about $T$ as a $K$ linear enomorphism of $K$ modules, and it wouldn't have much to say about the fact that $T$ is multiplicative.
