Lemma 10.97.6. Let $R$ be a Noetherian ring. Let $I$ be an ideal of $R$. The completion $R^\wedge$ of $R$ with respect to $I$ is Noetherian.

Proof. This is a consequence of Lemma 10.97.5. It can also be seen directly as follows. Choose generators $f_1, \ldots , f_ n$ of $I$. Consider the map

$R[[x_1, \ldots , x_ n]] \longrightarrow R^\wedge , \quad x_ i \longmapsto f_ i.$

This is a well defined and surjective ring map (details omitted). Since $R[[x_1, \ldots , x_ n]]$ is Noetherian (see Lemma 10.31.2) we win. $\square$

There are also:

• 5 comment(s) on Section 10.97: Completion for Noetherian rings

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).