**Yoshifumi Tsuchimoto**

Let be an abelian category. For any object of , the extension group is defined to be the derived functor of the ``hom'' functor

We note that the functor is a ``bifunctor''. We may thus consider the right derived functor of and that of . Fortunately, both coincide: The extension group may be calculated by using either an injective resolution of the second variable or a projective resoltuion of the first variable . See [1, Proposition 8.4,Corollary 8.5].

- We may compute them by using an injective resolution
- We may compute them by using a free resolution

In the last lecture we mentioned the notion of injective hulls. Although they are not essential part of our lecture, students may find it interesting to calculate some of the injective hulls of known modules. So we write down some definitions and results related to them.

- is an -submodule of which contains .
- is an essential extension of .

- The set has a maximal element.
- If is an injective -module, then any maximal element of is injective.

Injective hulls may then be used to obtain the ``minimal injective resolution'' of a module.