example of free module
from the definition, is http://planetmath.org/node/FreeModulefree as a -module for any positive integer .
A more interesting example is the following:
The set of rational numbers do not form a http://planetmath.org/node/FreeModulefree -module.
First note that any two elements in are -linearly dependent. If and , then . Since basis (http://planetmath.org/Basis) elements must be linearly independent, this shows that any basis must consist of only one element, say , with and relatively prime, and without loss of generality, . The -span of is the set of rational numbers of the form . I claim that is not in the set. If it were, then we would have for some , but this implies that which has no solutions for ,, giving us a contradiction. ∎
|Title||example of free module|
|Date of creation||2013-03-22 13:48:41|
|Last modified on||2013-03-22 13:48:41|
|Last modified by||mathcam (2727)|