The category of modules has enough injectives. Let $M$ be a module, and let $I^0$ be an injective module such that $$ 0 \longrightarrow M \longrightarrow I^0$$

is exact. Then, let $M_0$ be the image of $M$ in $I^0$ , and construct the factor module $I^0\slash M^0$ . Then, since the category of modules has enough injectives, we can find a module $I^1$ such that $$ 0 \longrightarrow I^0 \slash M^0 \stackrel{\phi_0}{\longrightarrow} I^1$$

is exact. $\phi_0$ induces a homomorphism $\phi \!:\! I^0 \longrightarrow I^1$ , whose kernel is $M^0$ . We thus have an exact sequence $$ 0 \longrightarrow M \longrightarrow I^0 \longrightarrow I^1.$$

One can continue this process to construct injective modules $I^n$ for any $n\in \mathbb Z$ (the resolution may terminate: $I^m = 0$ for some $N\in \mathbb Z$ with all $m > N$ ).