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

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Essential Extension - Properties

... is, if the module is essential in another module, then it is equal to that module) is an

... 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

... 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 -

... If f is a H-essential H-morphism with a domain X and an H-

**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 ...Main Site Subjects

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