One can define the K-theory space of a monoidal category $S$ in which every morphism is an isomorphism as the classifying space $B(S^{-1}S)$. Then we show that this definition coincides with the classical definition if $S=iP(R)$ the category of isomorphisms in $P(R)$. Here $P(R)$ is the category of finitely generated projective modules over a commutative unital ring $R$.

One way to do this is to consider $iF(R)$ the category of finitely generated free modules instead and then use a cofinality theorem. We can write the category $iF(R)$ as $$iF(R)=\coprod Gl_n(R),$$ where $Gl_n(R)$ is the category with one object and the general linear group as morphisms.

So far for the classical theory. I am considering the category $P(R,\mathbb G_m)$, the category of pairs $(P,\theta)$, where $P$ is finitely generated and projective and $\theta$ is an automorphism of $P$, morphisms have to respect the automorphisms. It would be desirable if I could write the category $iF(R,\mathbb G_m)$ in a similar fashion as $iF(R)$ above.

There are more objects but that's ok, what really gives me a headache are the morphisms. As far as I understand this there is a morphism $C:(R^n,A)\to (R^n,B)$ (now, $A,B$,and $C$ are matrices) if and only if $A$ and $B$ are similar and for any two morphisms $C,D$ we have $CD^{-1}\in Z(B)$ the centraliser of $B$.

So my question: are there any classification results for invertible matrices with entries in arbitrary rings? Sure, we have the characteristic polynomial but maybe there's more. Also is there a relation between the similarity class of a matrix and it's centraliser?

The question turned out to be pretty vague and I am sorry for that. My guess is that there are connections to representation theory. Any suggestion of literature is welcome, really.

Edit: The answers I got so far suggest to use the characteristic polynomials to reformulate the question as a question of modules of projective dimension 1 over the polynomial ring localised at the ideal of monic polynomials with constant part a unit (This is essentially what Grayson does in the paper K-theory of Endomorphisms). I tried this before I asked this question and it lead to a dead end, since the third part in the localisation sequence is as intractable as the one I am interested in. Another paper by Grayson (Weight filtrations via commuting automorphisms) suggests that the (reduced) K-theory space is the delooping of the (truncated ?) space for Karoubi-Villamayor K-theory. I am trying to prove this directly by using the universal property of higher K-theory (cf. group completion). For that I need to show that a certain map is a localisation on homology. My hope was that mimicking the proof for the classical Theorem (see above) would be helpful.