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I am aware of the theory of eigenvalues for matrices over fields. I was wondering to what extent this theory extends? Do we have a corresponding theory for matrices over integral domains, or at least over UFDs? H.C.Lee remarks here that there is no eigenvalue theory over general rings.

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    For a matrix $M\in M_n(A)$ where $A$ is a domain, the characteristic polynomial $P(M)$ has coefficients in $A$ and is monic. Thus, the roots of $P(M)$ all lie in an integral closure of $A$. Therefore, we can be sure that the roots belong to $A$ only if $A$ is integrally closed.2011-12-12

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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.