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

A module $Q$ is an injective module if it satisfies the following equivalent conditions:

(a) Every short exact sequence of the form $0 \to Q \to B \to C \to 0$ is split;

(b) The functor ${\rm Hom}(-, Q)$ is exact;

(c) If $f : X \to Y$ is a monomorphism and there exists a homomorphism $g : X \to Q$ , then there exists a homomorphism $h : Y \to Q$ such that $hf = g$ .

$\displaystyle \xymatrix{ 0 \ar[r] & X \ar[d]_g \ar[r]^f & Y \ar@{-->}[dl]^h \ & Q } $




"injective module" is owned by antizeus.
(view preamble | get metadata)

View style:


Attachments:
example of injective module (Example) by Glotzfrosch
product of injective modules is injective (Theorem) by joking
Log in to rate this entry.
(view current ratings)

Cross-references: monomorphism, functor, short exact sequence, equivalent, module
There are 7 references to this entry.

This is version 4 of injective module, born on 2001-12-12, modified 2004-03-11.
Object id is 1083, canonical name is InjectiveModule.
Accessed 4689 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)