The Stacks project

108.15 A Noetherian ring of infinite dimension

A Noetherian local ring has finite dimension as we saw in Algebra, Proposition 10.59.8. But there exist Noetherian rings of infinite dimension. See [Appendix, Example 1, Nagata].

Namely, let $k$ be a field, and consider the ring

\[ R = k[x_1, x_2, x_3, \ldots ]. \]

Let $\mathfrak p_ i = (x_{2^{i - 1}}, x_{2^{i - 1} + 1}, \ldots , x_{2^ i - 1})$ for $i = 1, 2, \ldots $ which are prime ideals of $R$. Let $S$ be the multiplicative subset

\[ S = \bigcap \nolimits _{i \geq 1} (R \setminus \mathfrak p_ i). \]

Consider the ring $A = S^{-1}R$. We claim that

  1. The maximal ideals of the ring $A$ are the ideals $\mathfrak m_ i = \mathfrak p_ iA$.

  2. We have $A_{\mathfrak m_ i} = R_{\mathfrak p_ i}$ which is a Noetherian local ring of dimension $2^ i$.

  3. The ring $A$ is Noetherian.

Hence it is clear that this is the example we are looking for. Details omitted.


Comments (0)


Post a comment

Your email address will not be published. Required fields are marked.

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).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 02JC. Beware of the difference between the letter 'O' and the digit '0'.