# Free modules

# Motivation

You often hear it said that a "free object" is an object with no other "relations" beyond those required of every object of that type, e.g., a "free group" is a group with no relations beyond those required of every group. But what does that actually mean, and how do you formalize it?

As a first attempt, we could start with a set

Let's try following this idea for modules, but this time filling in all of the details.

# The free module functor

## The goal

Let

Put more simply still, our functor

Before we construct the functor

Similarly, naturality "in ^{[1]}

This naturality condition will have many consequences for our construction, which we'll investigate later.

## The construction of

Let ^{[2]} In other words,

We then define a binary operation on ^{[3]}.

Finally, we let

In summary:

Given a set **free -module on ** is the set $$F(X)=\left{\sum_{x\in X}r_x x\mid r_x\in R,, r_x=0_R\text{ for all but finitely many }x\right}.$$

The group operation in

One more observation (to be elaborated upon later): there is a "copy"^{[4]} of the set

## Verifying the natural bijections

First suppose

We can verify this set map is bijective by constructing the inverse set map. Suppose

Can you verify our two set maps

# Free modules in general

Suppose **free** if **free on **.

In terms of elements, this is just like a basis for a vector space. If

For a more categorical approach, observe the following about subsets

- A subset
generates exactly when the corresponding module morphism is surjective; and - A subset
is "linearly independent in " exactly when the corresponding module morphism is injective. We can take this as the definition of linear independence, and declare the *relations*onto be the elements of the submodule .

## Suggested next note

If you're wondering why the vertical arrows in the second diagram are flipped, it's because a certain functor is "contravariant" (or equivalently, we need to use an "opposite" category as part of the formal formulation of what's going on). âŠī¸

Equivalently, we can define

to be the collection of all set maps with the property that for all but finitely many âŠī¸ This is precisely the construction of the free abelian group on the set

. Do you see how one could convert any finite formal sum of the given form into such a set map, and conversely? âŠī¸ More precisely, there is an injective set map from

to the underlying set of elements of , i.e., there is a set map . This is the unit of the adjunction. âŠī¸