Cokernel

The quotient of the codomain by the image of a module homomorphism.

Cokernel

Let $f:M\to N$ be a module homomorphism . The cokernel of $f$ is the quotient module

\operatorname{coker}(f)=N/\operatorname{im}(f),

where $\operatorname{im}(f)$ is the image of $f$ . It comes with a canonical surjection $N\to \operatorname{coker}(f)$ , and one always has an exact tail

M \xrightarrow{\,f\,} N \to \operatorname{coker}(f) \to 0.

Cokernels are the natural “targets” that make maps surjective by force, dual to how kernels make maps injective by force.

Examples:

For $f:\mathbb Z\to\mathbb Z$ given by $f(n)=kn$ with $k\ne 0$ , one has $\operatorname{coker}(f)\cong \mathbb Z/k\mathbb Z$ .
If $i:N\hookrightarrow M$ is the inclusion of a submodule, then $\operatorname{coker}(i)\cong M/N$ .
(Edge case) If $f$ is surjective, then $\operatorname{coker}(f)=0$ .