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

TL;DR

This paper characterizes the endomorphism ring of a specific product of p-adic modules, establishing an isomorphism, and provides methods for arithmetic operations and minimal polynomial analysis within this ring.

Contribution

It introduces an explicit ring isomorphism for the endomorphism ring and develops computational techniques for arithmetic and invertibility in this context.

Findings

Established a ring isomorphism to $E_{p,p^m}$.

Provided algorithms for computing inverses and negatives.

Characterized invertible elements and introduced minimal polynomials.

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.