The Stacks project

Lemma 10.107.4. If $A \to B \to C$ are ring maps and $A \to C$ is an epimorphism, so is $B \to C$.

Proof. Omitted. Hint: This is true in any category. $\square$

Comments (2)

Comment #5867 by Scott on

It's a bit confusing to say this is true in any category when it is explicitly specified that are ring maps (i.e., i assume, ring homomorphisms). Shouldn't this be stated in more general terms if it is true in any category? (Indeed I fail to see how it is true in any category as stated, since it is not given that is the same as . could be any arbitrary morphism from to ! I guess the real idea must be that if you can factorize the epimorohism through then the resulting morphism must also be epic?)

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