Lemma 10.107.8. If k \to S is an epimorphism and k is a field, then S = k or S = 0.
Proof. This is clear from the result of Lemma 10.107.7 (as any nonzero algebra over k is faithfully flat), or by arguing directly that R \to R \otimes _ k R cannot be surjective unless \dim _ k(R) \leq 1. \square
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