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Let $G_r=\mathrm{GL}_n(\mathbb{Z}/p^r)$. For $x\in G_r$ the centraliser $C_{G_{r}}(x)$ is abelian iff $x$ is regular iff the reduction mod $p$ of $x$ is regular. This is due to G. Hill, Regular elements and regular characters of $\mathrm{GL}_n(\mathcal{O})$, J. Algebra 174 (1995), no. 2, 610–635. The case $r=1$ is easy and was known earlier.