On the arithmetic of the endomorphism ring $\hbox {End}({\mathbb {Z}}_p[x]_{/\langle \overline{f}(x)\rangle }\times {\mathbb {Z}}_{p^2}[x]_{/\langle f(x)\rangle })$

Abstract

Let $p$ be a prime number, $f(x)$ a monic basic irreducible polynomial in $\mathbb {Z}_{p^2}[x]$ and $\overline{f}(x)=f(x)$ mod $p$. Set $F=\mathbb {Z}_p[x]_{/\langle \overline{f}(x)\rangle }$ and $R=\mathbb {Z}_{p^2}[x]_{/\langle f(x)\rangle }$, and denote by $\mathrm{End}(F\times R)$ the endomorphism ring of the $R$-module $F\times R$. We identify the elements of $\mathrm{End}(F\times R)$ with elements in a new set, denoted by $E_{p,f}$, of matrices of size $2\times 2$, whose elements in the first row belong to $F$ and the elements in the second row belong to $R$; also, using the arithmetic in $F$ and $R$, we introduce the arithmetic in $E_{p,f}$ and prove that $\mathrm{End}(F\times R)$ is isomorphic to the ring $E_{p,f}$. Moreover, we introduce the characteristic polynomial for each element in $E_{p,f}$ and consider its applications.