Short exact sequence

Remarks

Equivalently, the sequence is exact at $A$, at $B$ and at $C$. Equivalently again, $g \circ f = 0$, $f$ is a kernel of $g$, and $g$ is a cokernel of $f$. Then $C$ is isomorphic to the quotient $B / f(A)$, and $B$ is called an extension of $C$ by $A$.

Example: $0 \to \Z \xrightarrow{4} \Z \xrightarrow{\pi} \Z/4\Z \to 0$ is short exact, with $\pi$ the reduction modulo $4$. Multiplication by $4$ is injective, $\pi$ is onto, and the image $4\Z$ of the first map is the kernel of $\pi$.

Non-example: $0 \to 2\Z \xrightarrow{\iota} \Z \xrightarrow{\pi} \Z/4\Z \to 0$, with $\iota$ the inclusion, fails clause 3. Clauses 1 and 2 hold, but the image of $\iota$ is $2\Z$ while the kernel of $\pi$ is $4\Z$.