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

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.