Example 3

Let $m$ be a maximal ideal of $A$; the m-primary ideals are then the ideals $q$ of $A$ for which there exists an integer $n \geq 1$ such that $m^n \subset q \subset m$. For if $m^n \subset q \subset m$, $m$ is the only prime ideal containing $q$ (Chapter 11, $§ 1$, no. 1, Corollary to Proposition 1) and the conclusion follows from Example 2; conversely, if $q$ is m-primary, $m$ is the radical of $q$ and there therefore exists $n \geq 1$ such that $m^n \subset q$ (Chapter II, $§ 2$, no. 6, Proposition 15).