Eigenvalues for matrices over general rings

One generalization of the notion of eigenvalue can be found in module theory. One thinks of a linear operator $T : V \to V$ acting on a vector space $V$ as a module over the polynomial ring $k[T]$, and then an eigenvector $Tv = \lambda v$ spans a simple submodule of $V$. More generally, the generalized eigenspace associated to an eigenvalue $\lambda$ is an indecomposable submodule of $V$. The statement that $T$ has a Jordan normal form is then subsumed under the general theory of finitely-generated modules over principal ideal domains.

Generalizing, one may think of an $n \times n$ matrix over an arbitrary ring $R$ acting on column vectors over $R$ as describing an endomorphism of $R^n$ as a right $R$-module. This gives $R^n$ the additional structure of a left $R[T]$-module, and one can apply the general tools of module theory to study this module. Of course, if $R$ is complicated then the corresponding theory will be complicated.