Injective Hull - Generalization

Generalization

More generally, let C be an abelian category. An object E is an injective hull of an object M if M → E is an essential extension and E is an injective object.

If C is locally small, satisfies Grothendieck's axiom AB5) and has enough injectives, then every object in C has an injective hull (these three conditions are satisfied by the category of modules over a ring). Every object in a Grothendieck category has an injective hull.