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Let $f: M\to N$ be an $A$-module homomorphism . Establish 1-1 correspondence between these 2 sets :

$$\{\phi: K \to \text{Ker}(f) \mid \phi \text{ is an isomorphism}\}$$ and $$\{ g: K\to M \mid 0 \to K \to M \to N \text{ exact} \}$$

How to define the isomorphisms ?

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Let the first set be denoted by $\mathcal{A}$, the second by $\mathcal{B}$ and denote the inclusion $\operatorname{Ker}(f)\to M$ by $i$. Then for $\phi\in \mathcal{A}$ define $g:=i\phi$. This is in $\mathcal{B}$.

Now for the other direction let $g\in \mathcal{B}$. Then you can define $\phi:=i^{-1}g\in \mathcal{A}$ to be the image of $g$ under the inverse isomorphism.

Since it was only asked for the definition of the isomorphism I'll leave the part that these are well-defined and inverse to each other to the reader.

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