The Stacks project

Remark 10.28.13. Let $R$ be a ring. Let $\kappa $ be an infinite cardinal. By applying Example 10.28.6 and Proposition 10.28.8 we see that any ideal maximal with respect to the property of not being generated by $\kappa $ elements is prime. This result is not so useful because there exists a ring for which every prime ideal of $R$ can be generated by $\aleph _0$ elements, but some ideal cannot. Namely, let $k$ be a field, let $T$ be a set whose cardinality is greater than $\aleph _0$ and let

\[ R = k[\{ x_ n\} _{n \geq 1}, \{ z_{t, n}\} _{t \in T, n \geq 0}]/ (x_ n^2, z_{t, n}^2, x_ n z_{t, n} - z_{t, n - 1}) \]

This is a local ring with unique prime ideal $\mathfrak m = (x_ n)$. But the ideal $(z_{t, n})$ cannot be generated by countably many elements.

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