In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a free module is a
module
Module, modular and modularity may refer to the concept of modularity. They may also refer to:
Computing and engineering
* Modular design, the engineering discipline of designing complex devices using separately designed sub-components
* Modul ...
that has a
basis – that is, a
generating set consisting of
linearly independent elements. Every
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
is a free module, but, if the
ring of the coefficients is not a
division ring (not a
field in the
commutative case), then there exist non-free modules.
Given any
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
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 (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
s.
Definition
For a
ring and an
-
module
Module, modular and modularity may refer to the concept of modularity. They may also refer to:
Computing and engineering
* Modular design, the engineering discipline of designing complex devices using separately designed sub-components
* Modul ...
, 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, that is, for every subset
of distinct elements of
,
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.
[Proof: Suppose is free with a basis . For , must have the unique linear combination in terms of and , which is not true. Thus, since , there is only one basis element which must be a nonzerodivisor. The converse is clear.]
*If ''R'' is commutative, the polynomial ring