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*}

Example 10.34.7. Let $A$ be an infinite set. For each $\alpha \in A$, let $k_\alpha $ be a field. We claim that $R = \prod _{\alpha \in A} k_\alpha $ is Jacobson. First, note that any element $f \in R$ has the form $f = ue$, with $u \in R$ a unit and $e\in R$ an idempotent (left to the reader). Hence $D(f) = D(e)$, and $R_ f = R_ e = R/(1-e)$ is a quotient of $R$. Actually, any ring with this property is Jacobson. Namely, say $\mathfrak p \subset R$ is a prime ideal and $f \in R$, $f \not\in \mathfrak p$. We have to find a maximal ideal $\mathfrak m$ of $R$ such that $\mathfrak p \subset \mathfrak m$ and $f \not\in \mathfrak m$. Because $R_ f$ is a quotient of $R$ we see that any maximal ideal of $R_ f$ corresponds to a maximal ideal of $R$ not containing $f$. Hence the result follows by choosing a maximal ideal of $R_ f$ containing $\mathfrak p R_ f$.


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