The Stacks project

Lemma 10.107.3. If $R \to S$ is an epimorphism of rings and $R \to R'$ is any ring map, then $R' \to R' \otimes _ R S$ is an epimorphism.

Proof. Omitted. Hint: True in any category with pushouts. $\square$

Comments (2)

Comment #7337 by JS on

Unless I'm mistaken, this can be strengthened: if is an epimorphism of rings and is any ring map, then is an isomorphism. Proof: Use either the fact that if and are -modules and is a ring epimorphism, then the natural map is an isomorphism, or write

Comment #7338 by on

This is not true because in particular taking would give that and there are nontrivial epimorphisms of rings, for example .

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