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110.15 A Noetherian ring of infinite dimension

A Noetherian local ring has finite dimension as we saw in Algebra, Proposition 10.60.9. 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.

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