In abstract algebra, you learn about a variety of structures:
groups, rings, fields, vector spaces...
The study of vector spaces is often called “linear algebra.”
This subject has a wide range of applications.
But vector spaces can be generalized into something even more powerful -- the module.
The module is a generalization of a vector space.
So before we jump into the definition of a module,
let’s take a moment and remind ourselves what a vector space is.
It’s an abelian group V, and we call the elements of this group “vectors.”
It also comes with a field of scalars, F.
You can scale any vector 'v' by a scalar 'f', and the result is another vector.
You usually write the scalar on the left.
And you have a few standard rules from arithmetic:
the distributive properties, and the associative property.
But the key things to remember about vector spaces are these:
you have a group of vectors, and a field of scalars.
A module is defined the same way,
except instead of a field of scalars, you have a RING of scalars.
So if we go through and replace “F” with “R” -- “R” for ring --
and replace “V” with “M” -- “M” for module, you have the definition of a module.
It’s an abelian group “M” with a ring of scalars “R”.
For modules, you do NOT call the members of the group “vectors.”
That term is reserved for vector spaces.
Just use the word “elements” instead.
And notice that we multiply by the scalars on the left?
For this reason, you’ll hear some people call this a “left module.”
If you were to multiply by the scalars on the right, you would call it a “right module.”
But most of the time you’ll just hear the word “module.”
Some rings have an identity element, 1.
If the ring of scalars has an identity, we also require that 1 × m = m
for any element m in the module.
So a module is a vector space except the scalars are a ring.
Alternatively, you could say a vector space is a module where the scalars are a field.
Both statements are correct!
In fact, some people will learn about modules first, while others learn about vector spaces first.
But regardless of the path you took to this point,
it’s time to see some examples of modules,
and talk about the ways in which modules and vector spaces differ from one another.
For our first example, let M be ANY abelian group.
For the ring of scalars, let’s choose the integers, Z.
Here’s how we define scalar multiplication:
if 'r' is positive, then we define r × a to be (a +a + on and on) 'r' times.
If r is zero, then 0 × a = 0.
Be careful here.
This zero is from the scalar ring Z,
whereas THIS 0 is the identity element in the group M.
If you want to avoid confusion, you can always write a little subscript M on this 0.
And finally, if 'r' is negative, we define r × a to be (-a - a on and on)
the absolute value of R times.
The big takeaway here is that EVERY abelian group is also a Z-module.
For our next example, look at the 2-by-3 matrices where all the elements are integers.
We can make this into a left Z-module as follows:
define the multiplication of a matrix M by an integer 'r'
to be the matrix where every element is multiplied by 'r'.
Since we’re just beginning with modules,
let’s practice checking the axioms by seeing if the identity axiom holds.
If you multiply any matrix M by the identity “1”, we get this matrix.
But any integer multiplied by 1 is itself, so we get the same matrix.
So the identity axiom checks out.
Let’s now see a more complex example; one where the ring of scalars is NOT the integers.
For our group, let’s use R³, the group of 3-dimensional real vectors under addition...
And for the ring, we’ll use the three-by-three matrices with real number elements...
This ring is definitely not a field,
since not every square matrix has a multiplicative inverse.
And it’s not commutative, either.
But as a consolation prize, it does have an identity - the identity matrix.
For this module, we define the action of a matrix on a vector by matrix multiplication.
That is, we treat the scalar as linear map.
This example shows that modules can venture to places that vector spaces cannot go.
Just as groups can have subgroups, and rings can have subrings,
and vector spaces can have subspaces, modules can have submodules.
The idea and notation is the same.
If M is a module with scalars from a ring R,
then a submodule is a subgroup of M - let’s call it N - that is itself an R-module.
Here’s an example.
The multiples of three form a Z-module.
The multiples of six are a Z-SUBmodule.
The multiples of nine are another Z-submodule.
In fact, this module has an infinite number of submodules.
Now that we’ve seen some examples of modules,
let’s look at the differences between vector spaces and modules.
When you study vector spaces, there are two distinct categories:
finite dimensional & infinite dimensional vector spaces.
Finite dimensional vector spaces have a simple description,
and it’s completely determined by its dimension.
Suppose you have an N-dimensional vector space V.
Once you pick a basis,
every vector is represented by a tuple with 'n' components,
where each value is from the field of scalars F.
So an n-dimensional vector space is isomorphic to F^n … n-copies of F.
For modules, things are more complex.
Instead of talking about “finite dimensional spaces”,
we talk about finitely generated modules.
You CAN make a module that’s similar to a vector space;
a module that’s 'n' copies of the ring R.
We call this a “free module” because it’s free of the constraints and
complexities of other modules.
But you can also make modules that are finitely generated and are NOT simply copies of R.
For example, the direct product of Z, (Z mod 5) and (the multiples of 8)
is a finitely generated Z-module, and it’s NOT 3 copies of Z.
It’s not a free module.
This module departs from the simplicity of vector spaces.
Modules generalize the idea of a vector space.
By allowing the scalars to come from a ring instead of a field,
you get a more powerful tool, but the price you pay is a bit more complexity.
This largely comes from the fact that you can do things with rings
that you can’t do with fields.
For example, rings can have an ideal which allows you to make a quotient ring.
This is something you cannot do with a field.
And as we saw in our examples, you can use ideals and quotient rings to create modules
which are quite different from the flatter world of vector spaces.
If you’d like a hands-on demonstration of this structure, roll out some bread dough
and cut it into equal sized pieces.
You can use the integers as your ring of scalars.
To multiply a piece by 'n', just duplicate it 'n' times.
This is a great "roll module" for kids…