Let $K_ i$ be a family of fields. If $f = (f_ i) \in \prod K_ i$, then the ideal generated by $f$ is the same as the ideal generated by the idempotent $e = (e_ i)$ with $e_ i = 0, 1$ according to whether $f_ i$ is $0, 1$. Thus $D(f) = D(e)$ is open and closed and we conclude by Lemma 15.90.5 and Algebra, Lemma 10.25.5.
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