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I believe that the best way to think of it is in terms of modules. Say we have a matrix with characteristic polynomial $f$. For simplicity assume that its minimal polynomial is also $f$. Then it gives $R^n$ the structure of a $R[\alpha]/f(\alpha)$-module. This module is often locally free of rank $1$ because that property is preserved by specialization at primes1$, with different conditions depending on different ring properties.

So we are asked to understand line bundles on $\textrm{Spec} R[\alpha]/f(\alpha)$, or at least those line bundles that pushforward to trivial vector bundles on $\textrm{Spec} R$. This problem is quite subtle - it includes, for instance, computing the class numbers of every algebraic number field.
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I believe that the best way to think of it is in terms of modules. Say we have a matrix with characteristic polynomial $f$. For simplicity assume that its minimal polynomial is also $f$. Then it gives $R^n$ the structure of a $R[\alpha]/f(\alpha)$-module. This module is locally free of rank $1$ because that property is preserved by specialization at primes.

So we are asked to understand line bundles on $\textrm{Spec} R[\alpha]/f(\alpha)$, or at least those line bundles that pushforward to trivial vector bundles on $\textrm{Spec} R$. This problem is quite subtle - it includes, for instance, computing the class numbers of every algebraic number field.