The Stacks project

\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

Remark 10.27.12. Let $R$ be a ring. Let $\kappa $ be an infinite cardinal. By applying Example 10.27.6 and Proposition 10.27.7 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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