# Injective Hull

In mathematics, especially in the area of abstract algebra known as module theory, the injective hull (or injective envelope) of a module is both the smallest injective module containing it and the largest essential extension of it. Injective hulls were first described in (Eckmann & Schopf 1953), and are described in detail in the textbook (Lam 1999).

Read more about Injective Hull:  Definition, Examples, Properties, Uniform Dimension and Injective Modules, Generalization

### Other articles related to "injective, injective hull, injectives":

Essential Extension - Properties
... is, if the module is essential in another module, then it is equal to that module) is an injective module ... then possible to prove that every module M has a maximal essential extension E(M), called the injective hull of M ... The injective hull is necessarily an injective module, and is unique up to isomorphism ...
Injective Hull - Generalization
... An object E is an injective hull of an object M if M → E is an essential extension and E is an injective object ... 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 ...
Injective Module - Theory - Indecomposables
... Every injective submodule of an injective module is a direct summand, so it is important to understand indecomposable injective modules, (Lam 1999, §3 ... Every indecomposable injective module has a local endomorphism ring ... For an injective module M the following are equivalent M is indecomposable M is nonzero and is the injective hull of every nonzero submodule M is uniform M is the injective hull of a uniform module M is the ...
Injective Object - Injective Hull
... If f is a H-essential H-morphism with a domain X and an H-injective codomain G, G is called an H-injective hull of X ... This H-injective hull is then unique up to a canonical isomorphism ...