Lemma 10.107.1 (04VN)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 10: Commutative Algebra
Section 10.107: Epimorphisms of rings
Lemma 10.107.1 (cite)

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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.

Proof. Omitted. $\square$

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