example of free module

from the definition, n is http://planetmath.org/node/FreeModulefree as a -module for any positive integer n.

A more interesting example is the following:

Theorem 1.

The set of rational numbers Q do not form a http://planetmath.org/node/FreeModulefree Z-module.


First note that any two elements in are -linearly dependent. If x=p1q1 and y=p2q2, then q1p2x-q2p1y=0. Since basis (http://planetmath.org/Basis) elements must be linearly independent, this shows that any basis must consist of only one element, say pq, with p and q relatively prime, and without loss of generality, q>0. The -span of {pq} is the set of rational numbers of the form npq. I claim that 1q+1 is not in the set. If it were, then we would have npq=1q+1 for some n, but this implies that np=qq+1 which has no solutions for n,p ,q+, giving us a contradictionMathworldPlanetmathPlanetmath. ∎

Title example of free module
Canonical name ExampleOfFreeModule
Date of creation 2013-03-22 13:48:41
Last modified on 2013-03-22 13:48:41
Owner mathcam (2727)
Last modified by mathcam (2727)
Numerical id 5
Author mathcam (2727)
Entry type Example
Classification msc 13C10