Carnival time again
The 7th Carnival of Mathematics is up over at nOnoscience, including my latest discussion of knot coloring.
Some examples of modules
Today I want to run through a bunch of examples of the constructions we’ve been considering for modules. I’ll restrict to the case of a ring with unit.
One easy example of an -module that I’ve mentioned before is the ring
itself. We drop down to the underlying abelian group and then act on it using the ring multiplication. There are both left and right actions here:
and
where
and
are ring elements,
considered as an element of the module. We’ll start off by taking this module and sticking it into some of the constructions.
When we consider for some left
-module
the left module structures on
and
will get eaten and the right module structure on
will get flipped over, leaving us a left
-module. We can pick an element
by specifying
. Then
, telling us where everything else goes. If we write
for the homomorphism with
, then the left action of
on homomorphisms says
Thus . This means that
as left
-modules.
On the other hand, if we consider we get a right
-module. This consists of all
-linear functions from
to the ring
itself. We call this the “dual” module to
, and write
. Elements of the dual module are often called “linear functionals” on
.
Tensor products are even easier. When we consider for a left
-module
we can use the construction of tensor products to write an element as a finite sum:
. But then we can use the middle-linear property to write
, and then the linearity to collect all the terms together, giving
. The tensor product eats the module structure on
and the right module structure on
, leaving a left
-module structure. We calculate
so as left
-modules.
Now let’s take two left -modules
and
and make
. This is an abelian group — a
-module — as is
. Let’s write
as
as above and then tensor over
with
. Then we can compose homomorphisms
This is the “evaluation” homomorphism that takes an element and a homomorphism
and gives back
.
As a special case, we can take itself in place of
. We get an evaluation homomorphism
. This “canonical pairing” we often write as
for a linear functional
and module element
.
What if we composed with an element of instead of
? We use the evaluation homomorphism to get
So given a homomorphism we get a homomorphism
Of course, all this goes through suitably changed by swapping “right” for “left”. For example, given a right -module
we have a dual left
-module
.
What do we get if we start with a left module , dualize it, then dualize again to get another left module
? Following the definitions we see
. I claim that there is a natural morphism of left
-modules
. That is, a special element of
but we know that this is isomorphic to
which we write as
so we’re really looking for a special homomorphism from to
. And we’ve got one: the canonical pairing! So we take the canonical pairing as a homomorphism from
and pass it through this natural isomorphism to get a homomorphism
. In case this looks completely insane, here it is in terms of elements:
takes a linear functional
and gives back an element of the ring by the rule
.
Greatest Lower Bounds and Euclid’s Algorithm
One interesting question for any partial order is that of lower or upper bounds. Given a partial order and a subset
we say that
is a lower bound of
if
for every element
. Similarly, an upper bound
is one so that
for every
.
A given subset may have many different lower bounds. For example, if
is a lower bound for
and
, then the transitive property of
shows that
is also a lower bound. If we’re lucky, we can find a greatest lower bound. This is a lower bound
so that for any other lower bound
we have
. Then the set of all lower bounds is just the set of all elements below
.
As an explicit example, let’s consider the partial order we get from the divisibility relation on integers. In this case, a greatest lower bound is better known as a greatest common divisor, and for any pair of natural numbers we can find a greatest common divisor. The method we use actually goes all the way back to Euclid’s Elements, proposition VII.2. Because of this, it’s known as Euclid’s algorithm.
First we need remember how we divide numbers from back in arithmetic. Given two numbers and
we can find numbers
and
with
and
. We start with
and make up all the natural numbers of the form
with
a natural number. Since the natural numbers are well-ordered, there is a least element
in this set. Notice that here we use the regular order on
, not divisibility. Anyhow, I claim that
is less than
, so it will be our
. Indeed if it’s greater than
we can subtract
from it to get
: a lower natural number, which contradicts the fact that
is the lowest. Thus we have found
with
. Notice that
if and only if
.
Okay, now that we remember how to divide we’re ready for Euclid’s algorithm. We start with numbers and
and divide
into
to find
. If
then
and
is a common divisor. It’s clearly the greatest since any common divisor divides it.
If then I say that any common divisor of
and
is also a common divisor of
and
. We see that if
and
then
so
as well. Thus if we can find the greatest common divisor of
and
it will also be the greatest common divisor of
and
.
To do this, just keep going. Divide into
to get
. If
then we’re done, otherwise keep going again. At each step
, so we must eventually hit zero and stop. If we didn’t, then we could collect all the
together to have a set of natural numbers with no lowest element, contradicting the well-ordered property of
. At the end we’ll have the greatest common divisor of
and
.
In fact, if we watch closely we can do even better. We stop when with a remainder of zero, getting the greatest common divisor
. Now we look at the next-to-last step
. We can rewrite this as
. Then the step before this —
— can be written as
. This gives us
. We can keep doing this over and over, at each step writing
as a linear combination of the two remainders that we started a given step with. Taking this all the way back to the beginning of the algorithm, we get
for some integers
and
.
From here you should be able to see how to use Euclid’s algorithm to find the greatest common divisor of any finite set of natural numbers, and further how to write it as an integral linear combination of the elements of
.