Fundamental homomorphism theorem (epimorphism)

Question

Let φ : R → S be a ring epimorphism. Prove that R/kerφ ∼= S.

Is this the fundamental homomorphism theorem? I thought the FHT started with a ring homomorphism and not an epimorphism. Does this change the proof of the theorem ?

Answer 1

The general formula for the homomorphism theorem is $$R/\ker(\phi)\cong \mathrm{Im}(\phi)$$ In the special case, $\phi$ is an epimorphism, then $\mathrm{Im}(\phi)=S$, but be careful, you cannot deduce the general formula from the special case, for example when $\phi$ isn‘t an epimorphism

Answer 2

You can state the theorem without the epimorphism assumption replacing $S$ by $\mathrm{Im} \varphi$. So both formulation are equivalent.