Definition 1

Let $\mathbf{A}$ be a topological ring. If a two-sided ideal $m$ of $\mathbf{A}$ is such that the given topology on $\mathbf{A}$ is the $m$-adic topology, $m$ is called a defining ideal & the topology on $\mathbf{A}$.

Let $A$ be a commutative Noetherian ring, $m$ an ideal of $A$ and $t$ its radical (Chapter II, § 2, no. 6). If $m'$ is a defining ideal of the $m$-adic topology, there exists an integer $n > 0$ such that ${m'}^n \subset m$ ($§ 2$, no. 5) and hence $m' \subset t$; conversely, since $A$ is Noetherian, there exists an integer $k > 0$ such that $t^k \subset m$ (Chapter II, § 2, no. 6, Proposition 15) and hence $t$ is the largest defining ideal of the $m$-adic topology.