In $R\text{-}\mathbf{Mod}$, for $f : M \to N$:

• $\ker f = \{m \in M : f(m) = 0\}$ with the inclusion $\ker f \hookrightarrow M$.
• $\mathrm{coker}\, f = N / \mathrm{im}\, f$ with the projection $N \twoheadrightarrow N / \mathrm{im}\, f$.
• $\mathrm{im}\, f = f(M) \subseteq N$.

In $\mathbf{Ab}$, the kernel of $f : A \to B$ is the usual kernel subgroup $\ker f = f^{-1}(0)$. The cokernel is $B / f(A)$.

In $\mathbf{Vect}_k$, the kernel of a linear map $T : V \to W$ represented by a matrix $A$ is the null space $\ker A = \{v : Av = 0\}$. The cokernel is $W / \mathrm{col}(A)$, and $\mathrm{im}\, T = \mathrm{col}(A)$.

In $\mathbf{Ch}(\mathcal{A})$, the kernel of $f_\bullet : A_\bullet \to B_\bullet$ is computed degreewise: $(\ker f)^n = \ker(f^n : A^n \to B^n)$. The differentials on $\ker f$ are the restrictions of the differentials on $A_\bullet$.

In $\mathbf{Sh}(X)$, the cokernel of $f : \mathcal{F} \to \mathcal{G}$ is the sheafification of the presheaf $U \mapsto \mathcal{G}(U) / f(\mathcal{F}(U))$. The presheaf cokernel is not always a sheaf, so sheafification is necessary.

Poset categories (viewed as preadditive with trivial abelian group structure) generally do not have kernels or cokernels, since the zero morphism condition has no meaning. Kernels and cokernels are meaningful only in preadditive (enriched over Ab) categories.

In an abelian category, every monomorphism $\iota : K \rightarrowtail A$ is a kernel of some morphism — specifically, it is the kernel of its cokernel: $\iota = \ker(\mathrm{coker}\, \iota)$. Such monomorphisms are called normal. In $\mathbf{Grp}$, only inclusions of normal subgroups are normal monomorphisms.

For a linear map $T : V \to W$ in $\mathbf{Vect}_k^{\mathrm{fd}}$, $\dim V = \dim(\ker T) + \dim(\mathrm{im}\, T)$ and $\dim W = \dim(\mathrm{im}\, T) + \dim(\mathrm{coker}\, T)$. This gives $\dim V - \dim W = \dim(\ker T) - \dim(\mathrm{coker}\, T)$.

A functor $F : \mathcal{A} \to \mathcal{B}$ between abelian categories is left exact if it preserves kernels (equivalently, preserves all finite limits). It is right exact if it preserves cokernels. It is exact if it preserves both.

For any object $A$ in an abelian category, the functor $\mathrm{Hom}(A, -)$ is left exact: it preserves kernels but not necessarily cokernels. The failure of right exactness leads to the Ext functor as the right derived functor.

The functor $M \otimes_R -$ is right exact: it preserves cokernels but not necessarily kernels. The failure of left exactness leads to the Tor functor as the left derived functor.

In an abelian category, every morphism $f : A \to B$ factors as:

$A \xrightarrow{\pi} \mathrm{coim}\, f \xrightarrow{\sim} \mathrm{im}\, f \xrightarrow{\iota} B$

where $\pi$ is an epimorphism, the middle map is an isomorphism, and $\iota$ is a monomorphism. This is the canonical factorization of $f$.