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For a prime $p$, let$E_{p,p^m}=\{\begin{pmatrix}a&b\\p^{m-1}c&d\end{pmatrix}|a,b,c\in\mathbb{Z}_{p},~\mathrm{and}~d\in\mathbb{Z}_{p^{m}}\}$. We first establish a ring isomorphism from$\mathrm{End}(\mathbb{Z}_p\times\mathbb{Z}_p^m)$ onto $E_{p,p^m}$. We thenprovide the way to compute $-d$ and $d^{-1}$ using arithmetic in$\mathbb{Z}_{p}$ and $\mathbb{Z}_{p^{m}}$, and characterize invertible elementsin $E_{p,p^m}$. Moreover, we introduce the minimal polynomial for each elementin $E_{p,p^m}$ and given its applications.

On the arithmetic of the endomorphism ring End($\mathbb{Z}_{p}\times\mathbb{Z}_{p^{m}}$)