PlanetMath (more info)
 Math for the people, by the people.
Encyclopedia | Requests | Forums | Docs | Wiki | Random | RSS  
Login
create new user
name:
pass:
forget your password?
Main Menu
Owner confidence rating: High Entry average rating: No information on entry rating
injective hull (Definition)

Let $ X$ and $ Q$ be modules. We say that $ Q$ is an injective hull or injective envelope of $ X$ if $ Q$ is both an injective module and an essential extension of $ X$.

Equivalently, $ Q$ is an injective hull of $ X$ if $ Q$ is injective, and $ X$ is a submodule of $ Q$, and if $ g : X \to Q'$ is a monomorphism from $ X$ to an injective module $ Q'$, then there exists a monomorphism $ h : Q \to Q'$ such that $ h(x) = g(x)$ for all $ x \in X$.

$\displaystyle \begin{xy} *!C\xybox{ \xymatrix{ & 0 \ar[d] \ 0 \ar[r] & X \ar[r]^i \ar[d]_g & Q \ar@{-->}[dl]^h \ & Q' } } \end{xy}$

Every module $ X$ has an injective hull, which is unique up to isomorphism. The injective hull of $ X$ is sometimes denoted $ E(X)$.



"injective hull" is owned by mclase. [ full author list (2) | owner history (1) ]
(view preamble | get metadata)

View style:

Other names:  injective envelope
Log in to rate this entry.
(view current ratings)

Cross-references: monomorphism, submodule, injective, essential extension, injective module, modules
There are 3 references to this entry.

This is version 3 of injective hull, born on 2002-01-05, modified 2004-06-07.
Object id is 1378, canonical name is InjectiveHull.
Accessed 4917 times total.

Classification:
AMS MSC16D50 (Associative rings and algebras :: Modules, bimodules and ideals :: Injective modules, self-injective rings)

Pending Errata and Addenda
None.
[ View all 1 ]
Discussion
Style: Expand: Order:
forum policy

No messages.

Interact
post | correct | update request | add derivation | add example | add (any)