Deducing results in linear algebra from results in commutative algebra Here are two examples of results which can be deduced from commutative algebra: 


*

*Any $n\times n$ complex matrix is conjugate to a Jordan canonical matrix (can be proven using the structure theorem for modules over a PID, in this case $\mathbf{C}[T]$ - see for example these course notes).

*Commuting matrices have a common eigenvector (this can be seen as a consequence of Hilbert's Nullstellensatz, according to Wikipedia).


My question is, does anyone know of other examples of results in linear algebra which can be deduced (non-trivially*) from results in commutative algebra?
More precisely: Which results about modules over fields can be optained via modules over more general commutative rings? (Thanks to Martin's comment below for suggesting this precision of the question).
.* by "non-trivially," I mean you have to go deeper than simply applying module theory to modules over a field.

Criteria for unitarily similarity between two matrices is given by Specht in 1940 in a paper
titled “Zur Theorie der Matrizen” (“Toward Matrix Theory”). 
