Existence of injective hull Is that true that every module over a ring has an injective hull? 
The term "has an injective hull" appears in several different contexts, some of them say that modules over a ring has this property (by looking on Google) and specially on Wikipedia link we have the claim "Every module M has an injective hull. The dual notion of a projective cover does not always exist for a module, however a flat cover exists for every module." How to prove it? What is a reference with this Theorem?
 A: You could try to learn about injective hulls, for example, 


*

*Lam's Lectures on Modules and Rings starting around page 75. 

*Matsumura's Commutative ring theory page 281.

*Or in Goodearl's Introuction to Noncommutative Noetherian rings around page 94. 

*Or in Wisbauer's book around page 141. 

*Rowen's Ring theory book around page 218.

*Or around page 94 of Lambek's algebra book. 

*Anderson and Fuller's Rings and categories of modules, ( Theorem 18.10 page 207 Thanks Jack Schmidt!) 

*somewhere in Commutative Algebra: With a View Toward Algebraic Geometry

*Bourbaki books, but I can't lay my hand on pages atm.

*You can read about them also in D.G. Northcott's Homological algebra books starting page 37. 

*Grillet's algebra book somewhere

*You can also read about very interesting things concerning these topics in Faith's Rings and Things starting around page 54.
The Flat Cover Conjecture was no trivial thing, and you can read about it here.
The rings for which all right modules have a projective cover are called right perfect rings, which you can read about in Anderson and Fuller or Lam's First course in noncommutative rings.
