free module

Let R be a commutative ring with unity. A free moduleMathworldPlanetmathPlanetmath over R is a (unital) module isomorphicPlanetmathPlanetmathPlanetmath to a direct sumMathworldPlanetmathPlanetmathPlanetmathPlanetmath of copies of R. In particular, as every abelian groupMathworldPlanetmath is a -module, a free abelian group is a direct sum of copies of . This is equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath to saying that the module has a free basis, i.e. a set of elements with the that every element of the module can be uniquely expressed as an linear combinationMathworldPlanetmath over R of elements of the free basis. In the case that a free module over R is a sum of finitely many copies of R, then the number of copies is called the rank of the free module.

An alternative definition of a free module is via its universal propertyMathworldPlanetmath: Given a set X, the free R-module F(X) on the set X is equipped with a function i:XF(X) satisfying the property that for any other R-module A and any function f:XA, there exists a unique R-module map h:F(X)A such that (hi)=f.

Title free module
Canonical name FreeModule
Date of creation 2013-03-22 12:10:10
Last modified on 2013-03-22 12:10:10
Owner mathcam (2727)
Last modified by mathcam (2727)
Numerical id 9
Author mathcam (2727)
Entry type Definition
Classification msc 16D40
Related topic FreeGroup
Defines free module
Defines free abelian group
Defines free basis
Defines rank of a free module