Exercise A0.V.1.05

Proposition

The power set ring $\mathscr{P}(S)$ is Noetherian if and only if $S$ is finite.

Proof

Suppose $S$ is finite. Then every ideal of $\mathscr{P}(S)$ is finite as a set and thus finitely generated.

Conversely, suppose $\mathscr{P}(S)$ is Noetherian. Assume, for contradiction, that $S$ is infinite. Let $f : \mathbb{N} \to S$ be a sequence of distinct elements of $S.$ Let

$$I_n = \mathscr{P}\left(\left\{ f(1), \ldots , f(n) \right\}\right).$$

We get an infinite ascending chain of ideals $I_n \subseteq I_{n+1}.$ The chain never stabilizes, since $\{f(n+1)\} \not\in I_n$ and $\{f(n+1)\} \in I_{n+1}.$ This contradicts the hypothesis that $\mathscr{P}(S)$ is Noetherian.

$\blacksquare$