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 .
Given a set , we can construct a free -module over . The module is simply the direct sum of copies of , often denoted . We give a concrete realization of this direct sum, denoted by , as follows:
- Carrier: contains the functions such that for cofinitely many (all but finitely many) .
- Addition: for two elements , we define by .
- Inverse: for , we define by .
- Scalar multiplication: for , we define by .
A basis for is given by the set where
Define the mapping by . This mapping gives a bijection between and the basis vectors . We can thus identify these sets. Thus may be considered as a linearly independent basis for .
Many statements about free modules, which are wrong for general modules over rings, are still true for certain generalisations of free modules. Projective modules are direct summands of free modules, so one can choose an injection in a free module and use the basis of this one to prove something for the projective module. Even weaker generalisations are flat modules, which still have the property that tensoring with them preserves exact sequences, and torsion-free modules. If the ring has special properties, this hierarchy may collapse, e.g., for any perfect local Dedekind ring, every torsion-free module is flat, projective and free as well.
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- Govorov, V. E. (2001), "Free module", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4.