In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, a free module is a
module that has a ''basis'', that is, a
generating set that is
linearly independent. Every
vector space
In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...
is a free module, but, if the
ring of the coefficients is not a
division ring (not a
field in the
commutative
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Perhaps most familiar as a pr ...
case), then there exist non-free modules.
Given any
set and ring , there is a free -module with basis , which is called the ''free module on'' or ''module of formal'' -''linear combinations'' of the elements of .
A
free abelian group is precisely a free module over the ring
of
integer
An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
s.
Definition
For a
ring and 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
; and
*
is
linearly independent: for every set
of distinct elements,
implies that
(where
is the zero element of
and
is the zero element of
).
A free module is a module with a basis.
An immediate consequence of the second half of the definition is that the coefficients in the first half are unique for each element of ''M''.
If
has
invariant basis number, then by definition any two bases have the same cardinality. For example, nonzero commutative rings have invariant basis number. The cardinality of any (and therefore every) basis is called the rank of the free module
. If this cardinality is finite, the free module is said to be ''free of finite rank'', or ''free of rank'' if the rank is known to be .
Examples
Let ''R'' be a ring.
* ''R'' is a free module of rank one over itself (either as a left or right module); any unit element is a basis.
* More generally, If ''R'' is commutative, a nonzero ideal ''I'' of ''R'' is free if and only if it is a
principal ideal generated by a
nonzerodivisor, with a generator being a basis.
* Over a
principal ideal domain (e.g.,
), a submodule of a free module is free.
* If ''R'' is commutative, the polynomial ring