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We have a structure theorem for f.g. modules over $R$, whenever $R$ is a PID. In the case of $R=\mathbb{Z}/p\mathbb{Z}[x]$ ($p$ a prime), the structure theorem can be used to obtain the rational canonical form for matrices over the finite field $\mathbb{Z}/p\mathbb{Z}$.

I am interested in some kind of canonical form for matrices over $\mathbb{Z}/p^k\mathbb{Z}$. Is there such a canonical form in the literature?

This question naturally leads to a more concrete one: What is known about the structure of f.g. modules over $\mathbb{Z}/p^k\mathbb{Z}[x]$?

If I have a matrix $A \in \mathrm{Mat}_n(\mathbb{Z}/p^k\mathbb{Z})$, the relevant $\mathbb{Z}/p^k\mathbb{Z}[x]$-module is the ``vector space'' $(\mathbb{Z}/p^k\mathbb{Z})^n$, on which $x$ acts as multiplication by $A$.

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    $\begingroup$ A partial answer (cyclic modules killed by $p^2$) but which probably grasps part of the idea how to do more generally: mathoverflow.net/questions/268530/… $\endgroup$
    – YCor
    Commented Jan 31, 2018 at 12:41
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    $\begingroup$ You can lift $A$ to a matrix $B\in M_n(\mathbb{Z})$. Then you have $U,V\in GL_n(\mathbb{Z})$ such that $UBV$ is a Smith normal form. Since $U,V$ have determinant $\pm 1$, they still are invertible modulo $p^k$. By reduction mod $p^k$, there exists $U',V'\in GL_n(\mathbb{Z}/p^k\mathbb{Z})$ sucht that $U'AV'=diag (\bar{d}_1,\ldots,\bar{d}_s, \bar{0},\cdots, \bar{0})$, with $d_1\mid\cdots\mid d_s$. I kinda remember that in $\mathbb{Z}/p^k\mathbb{Z}$, we have $(\bar{a})=(\bar{b})\iff \bar{a}=\bar{u} \ \bar{b}, $ where $\bar{u}$ is a unit.Then maybe you can have some kind of uniqueness $\endgroup$
    – GreginGre
    Commented Jan 31, 2018 at 12:51

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The problem is open, and not because nobody tried. For instance, it is known that the number of similarity classes in $M_n(\mathbf Z/p^2 \mathbf Z)$ is equal to the number of simultaneous conjugacy classes of pairs of commuting matrices in $M_n(\mathbf Z/p\mathbf Z)$ (S. J AMBOR and W. PLESKEN , Normal forms for matrices over uniserial rings of length two, J. Algebra 358 (2012), 250–256). For small values of $n$, exact formulae are given in my paper in IUMJ Similarity of matrices over local rings of length two with Pooja Singla and Steven Spallone, which is also available on the arXiv.

From the classification/structure theorem point of view, these are wild problems, so you should not expect a good answer. You will find a discussion of the history and other results on this problem in the intro to my paper mentioned above.

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