The Stacks project

Example 10.35.7. 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 $\mathfrak p R_ f$.