Exact sequence in which $M/\operatorname{Ker}f$ and $\operatorname{Im}f$ appear

Question

Let $R$ be a ring and $M,N$ be $R$-module. Let $f:M→N$. I'm looking for an exact sequence in which $M,N,\operatorname{Im}f,M/\operatorname{Ker}f$ appear.

Do you have any good ideas? Thank you for your help.

Answer 1

The sequence $$0 \longrightarrow M/\text{ker}(f) \longrightarrow N \to \text{coker}(f) \to 0$$ is a short exact sequence, where $\text{coker}(f)=N/\text{im}(f)$.

Note that the map $M/\text{ker}(f) \longrightarrow N$ is monomorphism and the map $N \to \text{coker}(f)$ is epimorphism.