Editing Free module (section)

== Definition ==

For a [[ring (mathematics)|ring]] <math>R</math> and an <math>R</math>-[[module (mathematics)|module]] <math>M</math>, the set <math>E\subseteq M</math> is a basis for <math>M</math> if:
* <math>E</math> is a [[generating set of a module|generating set]] for <math>M</math>; that is to say, every element of <math>M</math> is a finite sum of elements of <math>E</math> multiplied by coefficients in <math>R</math>; and
* <math>E</math> is [[linearly independent]]: for every set <math>\{e_1,\dots,e_n\}\subset E</math> of distinct elements, <math>r_1 e_1 + r_2 e_2 + \cdots + r_n e_n = 0_M</math> implies that <math>r_1 = r_2 = \cdots = r_n = 0_R</math> (where <math>0_M</math> is the zero element of <math>M</math> and <math>0_R</math> is the zero element of <math>R</math>).

A free module is a module with a basis.<ref>{{cite book|author=Hazewinkel |title=Encyclopaedia of Mathematics, Volume 4|year=1989|url={{Google books|plainurl=y|id=s9F71NJxwzoC|page=110|text=A free module is a module with a basis}}|page=110}}</ref>

An immediate consequence of the second half of the definition is that the coefficients in the first half are unique for each element of ''M''.

If <math>R</math> has [[invariant basis number]], then by definition any two bases have the same cardinality. For example, nonzero commutative rings have invariant basis number. The cardinality of any (and therefore every) basis is called the '''rank''' of the free module <math>M</math>. If this cardinality is finite, the free module is said to be ''free of finite rank'', or ''free of rank'' {{mvar|n}} if the rank is known to be {{mvar|n}}.