A free module over a nonzero unit ring 
is a left module with a basis,
so that every element has a unique expression as a finite
linear combination of basis elements with coefficients
in 
. The standard example with a basis
of 
elements
is 
, the set of
all sequences 
with entries in 
. It has the following operations.
1. It is an additive Abelian group with respect to the componentwise sum of sequences,
 |
(1)
|
2. Any sequence can be multiplied by any element of 
according to the rule
 |
(2)
|
and this product satisfies the associative and distributive laws.
The modules isomorphic to 
for some nonnegative
integer 
are exactly the free modules with a finite basis.
For example, the quotient ring 
, where 
is an integer greater than 1,
is not free as a 
-module. It has 
elements, whereas a nonzero free
-module is
infinite and the zero free 
-module has only one element.
It is free as a module over itself.
A free module with a basis of 
elements can be constructed over
the ring 
from any set 
by 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 over 
. When 
is a field, 
is an abstract
vector space having the set 
as a vector basis.
Every module is the image of some free module under a module homomorphism.
Given a module 
generated by its subset 
, the map defined by
is 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 subset 
of 
is infinite. Take a set 
with a bijection
to 
and define 
as the free module with basis 
, 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)
|
If 
is a finite
set with 
elements, this module is 
after choosing an ordering of 
.
