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Let $p$ be a prime number and $M_2(\mathbb Z_p^t)$ the set of matrices of order $2$ over $\mathbb Z_p^t$. Let $A\in M_2(\mathbb Z_p^t)$ and $C(A)=\{X\in M_2(\mathbb Z_p^t): AX=XA\}$. What conditions we must impose to the matrix $A$ so that $C(A)$ coincides with the set of polynomials in $A$?

P.S. If $t=1$ then $\mathbb Z_p$ is a field and in this case the condition is that the minimum polynomial of $A$ coincides with the characteristic polynomial.

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