Lemma 10.107.1. Let $R \to S$ be a ring map. The following are equivalent
$R \to S$ is an epimorphism,
the two ring maps $S \to S \otimes _ R S$ are equal,
either of the ring maps $S \to S \otimes _ R S$ is an isomorphism, and
the ring map $S \otimes _ R S \to S$ is an isomorphism.