Corollary

Let $m$ be a maximal ideal of $\mathbf{A}$; for every integer $n > 0$, the only prime ideal containing $m^n$ is $m$.

Such an ideal $p$ must contain $m$ by Proposition 1 applied to $a_i = m$ for $1 \leq i \leq n$; as $m$ is maximal, $p = m$.