Let be a ring with 1 and let be a left unital -module.
- Show that is contained in an injective -module.
- Show that .
- Conclude that is contained in an injective module.
We know that every abelian group is contained in a divisible abelian group- so as a -module, is contained in an injective -module . Let be the inclusion map.
Now is an -bimodule and a -bimodule, a left -module, and is a left -module. Using the action described in this prior exercise, and are left -modules.
Moreover, as we argue, is a unital left -module via the action . Indeed, , so that is a -module homomorphism. Now , so that ; , so that ; , so that , and , so that .
Now every -module homomorphism is also an abelian group (hence -module) homomorphism; this gives an injective map . Note that for all -module homomorphisms and , , so that . In particular, is an -module homomorphism.
We also have . Thus map is clearly well defined. Moreover, if and , we have . So , and thus is an -module homomorphism. Now if , then since is injective; thus is injective.
So is isomorphically embedded in an injective -module.