Free module
Free modules are another instance of free objects. (see this post for a discussion on free objects and the construction of the free group.) They are the modules satisfying a universal property analogue to that of free groups. Here is a more constructive definition:
Definition: Let be a set and
be a ring. An
-module
is free on
if for all
there exist unique non-zero
and unique non-zero
such that
for some
The set
is said to be a basis or a set of free generators for
If
is commutative, we call
the rank of
Note that there is a natural injection identifying
with
This definition coincides with our intuition that the only relations among elements of should be the minimal relations required for
to be a module. For example, the integers modulo
,
is not a free
-module on any set. Indeed, although any element
can be uniquely written as an element of the underlying set, we have
so we don’t have unicity in the ring elements, as the definition requires.
To construct a free -module on a set
consider the free abelian group on
which you can obtain by quotienting the free group on
,
by the normal subgroup generated by the set of all elements of the form
in
(This will satisfy the analog universal property for abelian groups.) This group can be identified with
where the operation is still concatenation of words but written additively. We now naturally define the free -module over
by
with the -action defined as
This is easily seen to satisfy the defining properties of a module and the universal property of free modules:
Theorem: (Universal property) For
a set and
a unitary ring, the
-module
constructed above satisfy the universal property of free modules. In other words, for every
-module
and every map
there exists a unique
-module homomorphism
such that the following diagram commutes:
Proof: Let
be the natural inclusion
Define
on
as
Then, since
generates
extends uniquely by linearity to an
-module homomorphism on
and we have
as wanted.
QED
Note that as in the case of the free groups, this property uniquely characterizes free modules up to isomorphism.
Reference: D. S. Dummit, R. M. Foote, Abstract Algebra, 3rd ed.
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