Definition 1

Let $A$ be a ring, $F, G, H$ three right (resp. left) $A$-modules, $f$ a homomorphism of $F$ to $G$ and $g$ a homomorphism of $G$ to $H$. The ordered pair $(f, g)$ is called an exact sequence if $\overline{g}^{-1}(0) = f(F)$, that is if the kernel of $g$ is equal to the image of $f$.

The diagram

is also called an exact sequence.

Consider similarly a diagram consisting of four modules and three homomorphisms:

This diagram is said to be exact at $F$ if the diagram $E \xrightarrow{f} F \xrightarrow{g} G$ is an exact sequence; it is said to be exact at $G$ if $F \xrightarrow{g} G \xrightarrow{h} H$ is an exact sequence. If (5) is exact at $F$ and $G$, it is said to be exact or also an exact sequence. Exact sequences with any number of terms are similarly defined.

Recall the following results (loc. cit.), where $E, F, G$ denote right (resp. left)

A-modules, the arrows represent homomorphisms and $0$ denotes a module reduced to its identity element:

(a) To say that $0 \to E \xrightarrow{f} F$ is an exact sequence is equivalent to saying that $f$ is injective.

(b) To say that $E \xrightarrow{f} F \to 0$ is an exact sequence is equivalent to saying that $f$ is surjective.

(c) To say that $0 \to E \xrightarrow{f} F \to 0$ is an exact sequence is equivalent to saying that $f$ is bijective, that is that $f$ is an isomorphism of $E$ onto $F$.

(d) If $F$ is a submodule of $E$ and $i$ denotes the canonical injection of $F$ into $E$ and $p$ the canonical surjection of $E$ onto $E/F$, the diagram

is an exact sequence.

(e) Iff: $E \to F$ is a homomorphism, the diagram

(where $i$ is the canonical injection off $(0)$ into $E$ and $p$ the canonical projection of $F/f(E)$) is an exact sequence.

(f) For the diagram

to be an exact sequence, it is necessary and sufficient that there exist modules $S$, $T$ and homomorphisms $a : E \to S$, $b : S \to F$, $c : F \to T$ and $d : T \to G$ such that $f = b \circ a$, $g = d \circ c$ and the three sequences be exact.

Recall finally that iff: $E \to F$ is an A-module homomorphism, we set $\operatorname{Ker}(f) = f^{-1}(0)$, $\operatorname{Im}(f) = f(E)$, $\operatorname{Coim}(f) = E/f^{-1}(0)$ and $\operatorname{Coker}(f) = F/f(E)$. With this notation it is possible to take, in (9), $S = \operatorname{Im}(f) = \operatorname{Ker}(g)$ and $T = \operatorname{Im}(g)$ (canonically isomorphic to $\operatorname{Coker}(f)$).