The free module of rank over a nonzero unit ring , usually denoted , is the set of all sequences that can be formed by picking (not necessarily distinct) elements , , ..., in . The set is a particular example of the algebraic structure called a module since is satisfies the following properties.
1. It is an additive Abelian group with respect to the componentwise sum of sequences,
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2. One can multiply any sequence with any element of according to the rule
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and this product fulfils both the associative and the distributive law.
The term free module extends to all modules which are isomorphic to , i.e., which have essentially the same structure as . Note that not all modules are free. For example, the quotient ring , where is an integer greater than 1 is not free, since it is a -module having elements, and therefore it cannot be isomorphic to any of the modules , which are all infinite sets. Hence it is not free as a -module, while, of course, it is free as a module over itself.
A free module of rank can be constructed over the ring from any abstract set by simply taking all formal linear combinations of the elements of with coefficients in
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and defining the following addition
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and the multiplication
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The module thus obtained is often denoted by . It is generated by , which are independent objects: this explains why it deserves to be called free. In the particular case where is a field, is an abstract vector space having the set as a basis.
Free modules play a central role in algebra, since any module is the homomorphic image of some free module: given a module generated by its subset , the map defined by is evidently a surjective module homomorphism from to . This property can be generalized to all modules , since it is easy to make it work even if the generating set of is infinite: it suffices to take a set equipotent to , and to define as the free module of "infinite rank" formed by all linear combinations
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in which all but finitely many of the coefficients are equal to zero. The module is then isomorphic to the module direct sum
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Note that if is a finite set with elements, this module is precisely .