construction of an injective resolution

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 $I^{0}/M^{0}$ . Then, since the category of modules has enough injectives, we can find a module $I^{1}$ such that

0\\longrightarrow I^{0}/M^{0}\\lx@stackrel{{\\scriptstyle\\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 $m>N$ ).