Short Exact Sequence
An exact sequence which contains three modules, with zeros on either side
is called a short exact sequence.
In the short exact sequence as written above, \(f\) is injective and \(g\) is surjective.
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.
If \(\{0\} \to A \xrightarrow{f} B \xrightarrow{g} C \to \{0\}\) is a short exact sequence, we say that \(B\) is an extension of \(C\) by \(A\).
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