Short Exact Sequence

Proof

The map on the far left maps zero to zero, while the map on the far right maps everything to zero. This means that $\ker (f)=\{0\}$ and $f$ is injective, as $\{0\}$ is the image of the zero map, while $\mathrm{im}(g)=C$ and $g$ is surjective, as $C$ is the kernel of the zero map.

This language is best understood by the relationship to quotient modules. $f$ can be thought of as providing an isomorphism between $A$ and a submodule of $B$, while surjectivity of $g$ means that, by the first isomorphism theorem that

In this sense every short exact sequence

leads to a quotient module isomorphism

and every quotient module $B/A$ leads to a short exact sequence