Let $A$ be a Noetherian ring, $\mathfrak{m}\subseteq A$ a maximal ideal, and $\mathfrak{q}\subseteq A$ any other proper ideal. The following are equivalent:

1. $\mathfrak{q}$ is $\mathfrak{m}$-primary.
2. $r(\mathfrak{q})=\mathfrak{m}$ (by 5. Proposition 4.9.2 - m-primary ideals)
3. There exists $n\in \mathbb{N}$ such that 1;;${\mathfrak{m}}^n\subseteq \mathfrak{q}\subseteq \mathfrak{m}$