Corollary 2

Let $A$ be a Noetherian ring and $m$ an ideal of $A$. The following conditions are equivalent:

(a) $A$ is a semi-local ring and $m$ is a defining ideal of $A$. (b) $A$ is a Zariski ring with the $m$-adic topology and $A/m$ is Artinian.

If (a) holds, $A$ is a Zariski ring with the $m$-adic topology (Chapter III, § 3, no. 3, Example 3); further, as by hypothesis $m$ contains a power of the Jacobson radical $r$ of $A$, every prime ideal of $A$ which contains $m$ also contains $r$ (Chapter II, § 1, no. 1, Proposition 1); it is therefore maximal, since $r$ is a finite intersection of maximal ideals (\emph{loc. cit.}); Proposition 9 then shows that $A/m$ is Artinian. Conversely, if (b) holds, every maximal ideal $p$ of $A$ contains the Jacobson radical of $A$ and hence contains $m$ (Chapter III, § 3, no. 3, Proposition 6); as $A/m$ is Artinian, the ideals $p/m$ are finite in number (Proposition 9) and hence $A$ has only a finite number of maximal ideals, which implies that it is semi-local.