Definition
A free module is a module with a basis: a linearly independent generating set.
For an -module, the set is a basis for if:
- is a generating set for ; that is to say, every element of is a finite sum of elements of multiplied by coefficients in ;
- is linearly independent, that is, if for distinct elements of, then (where is the zero element of and is the zero element of ).
If has invariant basis number, then by definition any two bases have the same cardinality. The cardinality of any (and therefore every) basis is called the rank of the free module, and is said to be free of rank n, or simply free of finite rank if the cardinality is finite.
Note that an immediate corollary of (2) is that the coefficients in (1) are unique for each .
The definition of an infinite free basis is similar, except that will have infinitely many elements. However the sum must still be finite, and thus for any particular only finitely many of the elements of are involved.
In the case of an infinite basis, the rank of is the cardinality of .
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