Proposition 3

(a) For

to be an exact sequence, it is necessary and sufficient that $f$ be injective.

(b) For

to be an exact sequence, it is necessary and sufficient that $f$ be surjective.

(c) For

to be an exact sequence, it is necessary and sufficient that $f$ be bijective (in other words (no. 2, Proposition 2) that $f$ be an isomorphism of E onto F).

(d) If F is a submodule of E and $i : F \to E$ is the canonical injection, $p : E \to E/F$ the canonical homomorphism, the diagram (14)

is an exact sequence.

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

(where $i$ is the canonical injection and $p$ the canonical surjection) is an exact sequence.

The proposition follows immediately from the definitions and Proposition 2 of no. 2.