On the Arithmetic of Endomorphism Ring End$$({Z_p} \times {Z_{{p^m}}})$$

For a prime p, let \({E_{P,{P^m}}} = \left\{ {\left( {\begin{array}{*{20}{c}}aamp;b \\ {{p^{m – 1}}c}amp;d \end{array}} \right)\left| {a,b,c \in {Z_p},d} \right. \in {Z_{{p^m}}}} \right\}\). We first establish a ring isomorphism from \({Z_{p,{p^m}}}\)onto \({E_{p,{p^m}}}\). Then we provide a way to compute -d and d^–1 by using arithmetic in Z_p and \({Z_{{p^m}}}\), and characterize the invertible elements of \({E_{p,{p^m}}}\). Moreover, we introduce the minimal polynomial for each element in \({E_{p,{p^m}}}\)and give its applications.