Definition

Let be a set. A free -module is a R-module with a function (of sets)

such that for all -modules and functions

there exists a unique -module homomorphism

with makes the following diagram commute:

Uniqueness

If they exist, free -modules are unique (up to isomorphism).

Proof

Let and are free -modules on . Then we have the diagram by stacking diagrams from above

Then by uniqueness and thus is an isomorphism.

Relation to direct sums

The direct sum is a free -module on .

Proof

Let

where is the element in the ""th position even though they aren’t ordered. It is element with associated with the element and elsewhere.

Given a set map for an -module , let . By linearity we have

which determines the function which is a homomorphism of -modules.

Notes

For finite free modules and we can write the generators out, then using the universal property we get the following relationship:

Thus, free modules are anologues of vector spaces. They have finite number of “basis” elements and maps between them are completely determined by these basis which can be written (unique for a choice of basis) as a matrix.

Relation to generating sets

A module is generated by if and only if there exists a surjective module homomorphism

Looking at free modules this means is finitely generated if and only if there is a surjective module homomorphism

Therefore, finitely generated means that the module is a quotient of .