projective module

A module P is projective if it satisfies the following equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath conditions:

(a) Every short exact sequenceMathworldPlanetmathPlanetmath of the form 0ABP0 is split (;

(b) The functorMathworldPlanetmath Hom(P,-) is exact (;

(c) If f:XY is an epimorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath and there exists a homomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmath g:PY, then there exists a homomorphism h:PX such that fh=g.


(d) The module P is a direct summandMathworldPlanetmath of a free moduleMathworldPlanetmathPlanetmath.

Title projective moduleMathworldPlanetmath
Canonical name ProjectiveModule
Date of creation 2013-03-22 12:09:42
Last modified on 2013-03-22 12:09:42
Owner antizeus (11)
Last modified by antizeus (11)
Numerical id 7
Author antizeus (11)
Entry type Definition
Classification msc 16D40
Related topic InvertibleIdealsAreProjective