Exercise 4, § 3

Let $A$ be a semi-local ring and $r$ its Jacobson radical. For $A$ to be Noetherian, it is necessary and sufficient that every ideal of $A$ be closed under the $r$-adic topology and that every maximal ideal of $A$ be finitely generated. (If these conditions hold, show first that the Hausdorff completion $\hat{A}$ of $A$ is Noetherian, using § 2, no. 13, Corollary to Proposition 19 and no. 10, Corollary 5 to Theorem 1. Then observe that the $r$-adic topology is Hausdorff on $A$ and that there exists an increasing injection of the set of ideals of $A$ into the set of ideals of $\hat{A}$.)