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The Fundamental Homomorphism Theorem states that:

Let R, R' be rings, ϕ : R → R' a homomorphism. Then with K = kerϕ, there is an isomorphism between R/K and ϕ(R).

However in some example questions i have seen, they prove that the function is not only a homomorphism but surjective also. Is proving surjectivity also necessary for this theorem?

cf12418
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    Surjectivity allows you to replace $\phi\left(R\right)$ by $R^{\prime}$ (which is probably what they do in said examples). – darij grinberg May 25 '14 at 18:42

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