Lemma 10.107.1. Let $R \to S$ be a ring map. The following are equivalent

1. $R \to S$ is an epimorphism,

2. the two ring maps $S \to S \otimes _ R S$ are equal,

3. either of the ring maps $S \to S \otimes _ R S$ is an isomorphism, and

4. the ring map $S \otimes _ R S \to S$ is an isomorphism.

Proof. Omitted. $\square$

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).