Lemma 10.106.4. 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. $\square$

Comment #5867 by Scott on

It's a bit confusing to say this is true in any category when it is explicitly specified that $A \rightarrow B \rightarrow C$ 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 $A \rightarrow C$ is the same as $A \rightarrow B \rightarrow C$. $A \rightarrow C$ could be any arbitrary morphism from $A$ to $C$! I guess the real idea must be that if you can factorize the epimorohism $A \rightarrow C$ through $B$ then the resulting morphism $B \rightarrow C$ must also be epic?)

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