# 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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