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,
 |
(1)
|
2. One can multiply any sequence with any element of 
according to the rule
 |
(2)
|
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 
 |
(3)
|
and defining the following addition
 |
(4)
|
and the multiplication
 |
(5)
|
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
 |
(6)
|
in which all but finitely many of the coefficients 
are equal to zero. The module 
is then isomorphic to the module
direct sum
 |
(7)
|
Note that if 
is a finite set with 
elements, this module is precisely 
.
