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**NonCommAlg** every simple module over a commutative ring $\displaystyle R$ is isomorphic with $\displaystyle R/I,$ where $\displaystyle I$ is some maximal ideal of $\displaystyle R.$ now your ring has only one maximal ideal $\displaystyle M$ and so if $\displaystyle N$ is a simple $\displaystyle R$ module, then

$\displaystyle N \cong R/M.$ In particular $\displaystyle |N|=|R/M|=p^n.$ now let $\displaystyle (0)=M_0 < M_1 < \cdots < M_{m-1}=M < M_m=R$ be a composition series for $\displaystyle R.$ then $\displaystyle |M_i/M_{i-1}|=p^n,$ for all $\displaystyle 1 \leq i \leq m,$ because every

$\displaystyle M_i/M_{i-1}$ is a simple $\displaystyle R$ module. therefore $\displaystyle |R|=|M_m|=\prod_{i=1}^m |M_i|/|M_{i-1}|=\prod_{i=1}^m |M_i/M_{i-1}|=\prod_{i=1}^m p^n=p^{mn}.$