Remark

(2) To say that there is an exact sequence

means that $f$ is injective, $g$ surjective and that the canonical bijection associated with $g$ is an isomorphism of $F/f(E)$ onto $G$. It is also said that the triple $(F, f, g)$ is an extension of the module G by the module E (I, § 6, no. 7).

(3) If there is an exact sequence with 4 terms

the cokernel of $f$ is $F/f(E) = F/g^{-1}(0)$ and the kernel of $h$ is $g(F)$; the canonical bijection associated with $g$ is therefore an isomorphism

(4) Consider an ordered pair of A-module homomorphisms

For diagram (15) to be an exact sequence, it is necessary and sufficient that there exist two A-modules S, T and homomorphisms $a : E \to S,\ b : S \to F,\ c : F \to T,\ d : T \to G$ such that the three sequences

are exact and $f = b \circ a$ and $g = d \circ c$.

If (15) is an exact sequence, then take $S = f(E) = g^{-1}(0)$ and $T = g(F)$, $b$ and $d$ being the canonical injections and $a$ (resp. $c$) the homomorphism with the same graph as $f$ (resp. $g$). Conversely, if S, T, a, b, c, d satisfy the above conditions, then $f(E) = b(a(E)) = b(S)$ and $g^{-1}(0) = c(d(0)) = c(0)$, hence the exactness of (16) shows that $f(E) = g^{-1}(0)$.

The use of explicit letters to denote homomorphisms in an exact sequence is often dispensed with when it is not necessary for the arguments.