injective module


A module Q is an injective moduleMathworldPlanetmath if it satisfies the following equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath conditions:

(a) Every short exact sequenceMathworldPlanetmathPlanetmath of the form 0QBC0 is split (http://planetmath.org/SplitShortExactSequence);

(b) The functorMathworldPlanetmath Hom(-,Q) is exact (http://planetmath.org/ExactFunctor);

(c) If f:XY is a monomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath and there exists a homomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath g:XQ, then there exists a homomorphism h:YQ such that hf=g.

\xymatrix0\ar[r]&X\ar[d]g\ar[r]f&Y\ar@-->[dl]h&Q
Title injective module
Canonical name InjectiveModule
Date of creation 2013-03-22 12:02:26
Last modified on 2013-03-22 12:02:26
Owner antizeus (11)
Last modified by antizeus (11)
Numerical id 8
Author antizeus (11)
Entry type Definition
Classification msc 16D50