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

Abstract

For a prime $p$, let $E_{p,p^m}=\{\left(\begin{array}{cc}a& b\\ p^{m-1}c& d\end{array}\right)|a,b,c\in {\mathbb{Z}}_p,\mathrm{and}d\in {\mathbb{Z}}_{p^m}\}$. 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.

12 pages