The sequence $0 \to A \xrightarrow{f} B$ is exact at $A$ if and only if $f$ is a monomorphism ($\ker f = 0$). Saying "the kernel is zero" means $f$ is injective in $R\text{-}\mathbf{Mod}$.

The sequence $A \xrightarrow{f} B \to 0$ is exact at $B$ if and only if $f$ is an epimorphism ($\mathrm{im}\, f = B$). In $R\text{-}\mathbf{Mod}$, this means $f$ is surjective.

For any morphism $f : A \to B$ in $R\text{-}\mathbf{Mod}$, the sequence

$0 \to \ker f \xrightarrow{\iota} A \xrightarrow{f} B \xrightarrow{\pi} \mathrm{coker}\, f \to 0$

is exact. The first isomorphism theorem states $A / \ker f \cong \mathrm{im}\, f$.

For $N \trianglelefteq G$ in $\mathbf{Grp}$, the sequence $1 \to N \xrightarrow{\iota} G \xrightarrow{\pi} G/N \to 1$ is exact (where $1$ denotes the trivial group). In $\mathbf{Ab}$, this becomes $0 \to N \to G \to G/N \to 0$.

For a Noetherian ring $R$ and a multiplicative set $S$, the functor $S^{-1}(-)$ is exact: it preserves exact sequences. In particular, $0 \to S^{-1}M' \to S^{-1}M \to S^{-1}M'' \to 0$ is exact whenever $0 \to M' \to M \to M'' \to 0$ is.

A short exact sequence of chain complexes $0 \to A_\bullet \to B_\bullet \to C_\bullet \to 0$ induces a long exact sequence in homology:

$\cdots \to H_n(A) \to H_n(B) \to H_n(C) \xrightarrow{\delta} H_{n-1}(A) \to H_{n-1}(B) \to \cdots$

The connecting homomorphism $\delta$ is constructed via the Snake Lemma.

For an open cover $X = U \cup V$, there is a long exact sequence

$\cdots \to H_n(U \cap V) \to H_n(U) \oplus H_n(V) \to H_n(X) \xrightarrow{\delta} H_{n-1}(U \cap V) \to \cdots$

This is the Mayer-Vietoris sequence, a fundamental computational tool in algebraic topology.

For $R$-modules, a short exact sequence $0 \to A \to B \to C \to 0$ induces a long exact sequence

$0 \to \mathrm{Hom}(C, M) \to \mathrm{Hom}(B, M) \to \mathrm{Hom}(A, M) \to \mathrm{Ext}^1(C, M) \to \mathrm{Ext}^1(B, M) \to \cdots$

This is the long exact sequence for Ext.

A functor $F : \mathcal{A} \to \mathcal{B}$ between abelian categories is exact if it sends every short exact sequence $0 \to A \to B \to C \to 0$ to a short exact sequence $0 \to FA \to FB \to FC \to 0$. Localization, direct limit, and base change (for flat modules) are exact functors.

$F$ is left exact if it sends $0 \to A \to B \to C \to 0$ to the exact sequence $0 \to FA \to FB \to FC$. The functor $\mathrm{Hom}(M, -)$ is left exact; its right derived functors are $\mathrm{Ext}^n(M, -)$.

$F$ is right exact if it sends $0 \to A \to B \to C \to 0$ to $FA \to FB \to FC \to 0$. The functor $M \otimes_R -$ is right exact; its left derived functors are $\mathrm{Tor}_n(M, -)$.

The global sections functor $\Gamma(X, -) : \mathbf{Sh}(X) \to \mathbf{Ab}$ is left exact but not exact in general. Its right derived functors are the sheaf cohomology groups $H^n(X, -)$.