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

Abstract: 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 then provide the way to compute $-d$ and $d^{-1}$ using arithmetic in $\mathbb{Z}{p}$ and $\mathbb{Z}{p^{m}}$, and characterize invertible elements in $E_{p,p^m}$. Moreover, we introduce the minimal polynomial for each element in $E_{p,p^m}$ and given its applications.