Lemma 10.49.4. Let $k$ be a field. Let $S$ be a geometrically integral $k$-algebra. Let $R$ be a $k$-algebra and an integral domain. Then $R \otimes _ k S$ is an integral domain.

Proof. By Lemma 10.43.5 the ring $R \otimes _ k S$ is reduced and by Lemma 10.47.7 the ring $R \otimes _ k S$ is irreducible (the spectrum has just one irreducible component), so $R \otimes _ k S$ is an integral domain. $\square$

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