Ring theory is namely a branch of algebra that studies a
type of algebraic structure called ring. A ring is built
upon an abelian group defining the addition, and adds
multiplication which is another binary operation that is
closed, commutative and distributive over the addition.
The multiplication is not limited to something originated
from repetive sum of the addition, things like convolution
that has distributive properties can be used in the place
of multiplication. We will see how the internal abelian
group and the distributive property affects the behaviour
of ring during the studying of ring theory.
I find it nominating different names from time to time
when studying ring theory. So in case of our getting lost,
in this text, we aim at introducing and getting used to
basic and common concepts of rings.
Rings
A ring is a set equipped with
two binary operations addition and multiplication
, such that:
- is an abelian group with identity
called zero.
- is a semigroup, which is an
algebraic structure requiring:
- (Closed) .
- (Associative) ).
- (Distributive) Multiplication is distributive over
addition, which means :
- .
- .
A ring is called a ring with unity when
is a monoid, which adds a third
rule of
to the semigroup. The multiplicative identity
is called the unity of the ring.
The unity of the ring is unique, which can be shown
by assuming there's another unity
but .
A ring is called a commutative ring if the
multiplication is also commutative that
.
For convenience, we use a notation that
, and
for adding by itself for times,
for multiplying by itself for times,
for . It won't be hard
to notice that there's by
applying the distributive rule in a ring with unity,
and let's define for the
ring with unity.
For any ring, there's
:
consider
,
by distributive rule we have
,
and by subtracting from both
sides we have . The
can be proved analogously.
For any ring, there's also
, this can
be done by subtracting from each sides of
.
There's also directly
following this property.
With the set and
we can define the smallest ring possible, and such
a ring is also called the zero ring. It's
also the smallest ring with unity by setting
. However, in any set
with element other than , if we
permit then there's
,
which is a contradiction.
Just like how we can define a subgroup of a group,
we can define a subring of a ring to be a
algebraic substructure satisfying the ring axioms,
and denote it as . For any ring ,
there're and
, which are called the trivial subring.
Zero divisors and units
The discussion of a ring starts from basically around
the zero and unity .
The left zero divisors are elements
such that for some non-zero element
,
there's . Analogously
we define the right zero divisors and two-sided
zero divisors. We simply refer to zero
divisors when we don't care whether it's
placed on the left or the right. Obviously
is trivially a two-sided zero
divisor by
.
If a ring has no non-trivial zero divisor, it's
called a domain. Being a domain means
when , there's either
or .
Conventionally, we denote an abstract domain as
, to distinguish a proposition or discussion
of domains from those of rings.
We can easily find some examples of non-trivial
zero divisors, like and in the ring
due to
. And in the ring
like where is
prime, and , there's only trivial
zero divisor.
The units of ring with unity
are invertible elements such that
or
.
Being a unit and being a non-trivial zero divisor
are mutually exclusive: assume is a unit
and a non-trivial zero divisor such that
,
then by
,
it is a contradiction. So assume
is a unit but
, then by
,
the inverse of is unique. In this way, all
units in the ring with unity forms a group
called the unit group, denoted as .
In a non-zero ring, including in the
unit group will surely violate the bijectivity of
multiplication. A ring with unity such that all
non-zero elements are invertible is called a
division ring. And since units are not
non-trivial zero divisors, a division ring is
undoubtedly a domain.
By adding the commutativity to domains with unity
and division rings, we derive the definition of
integral domains and fields.
We can also easily find some example of fields,
like where is prime
is a finite field, ,
and are infinite fields
(you can quickly verify the addition, multiplication,
existence of unity and invertibility yourself).
For the example of integral domain that is not
a field, consider : it's a domain and
commutative so it's an integral domain, but it's
not a field since no element except
is invertible.
For the example of division ring that is not a
field, take the quaternions as an
example: a quaternion is defined as linear
combination of basis in the form of
where ,
,
and
.
The multiplications of the basis
pairwisely
are sufficiently examples that is
not commutative. But for any quaternion, we can
see that:
For any non-zero quaternion we have
, and
,
thus every non-zero quaternion is invertible.
And quaterinons form a division
ring that is not a field.
Characteristic of rings
The characteristic of ring , denoted
as , is the smallest
positive number such that
. If such
number does not exist (like there's element
of infinite order in the additive group), we
define to be .
The finite order of unity is the characteristic
Let the order of be in the
ring of unity , if ,
by
it violates the definition of characteristic.
So necessarily there must be
.
However, take arbitrary , there's
.
So is sufficiently a number that fulfils
. When combined
with the necessarity that
, is the smallest
number that fulfils the requirement and is the
characteristic of the ring.
Domains with unity are of zero or prime characteristic
We've seen domains with unity that are of zero
characteristic, like , ,
and . So we just need
to consider the cases of non-zero characteristic.
Assume a domain with unity is of
characteristic that is composite and can be
factorized as . Consider the sum of
unities, by distributive rule we have
.
Since there's no non-trivial zero divisor in ,
either or , and
both are smaller than . So the order of
is smaller than , so does the
characteristic of , which is a contradiction.
So domain with unity must have zero or prime
characteristic, so do the integral domains,
division rings and fields.
Freshman's dream of commutative rings of prime characteristic
When the ring is commutative, it satisfies the
binomial theorem that
.
This can be easily proved by inspecting how the
expansion of polynomial can be
applicable to the case of commutative ring.
However, the binomial theorem is not necessarily
applicable to the case of non-commutative rings,
which is expanded into
instead. Each term is a string of length
with alphabets , and there's no way
to ensure there're mergable terms among them.
When the commutative ring is of prime
characteristic , consider the coefficients
in , which are in the form of
:
is divisible by , while every factor of
and is smaller than and thus
not divisible by when . So
divides when , terms
except and are evaluated to
and we have
for commutative ring of prime characteristic.
This is also called the "freshman's dream" of
commutative rings of prime characteristic.
Wedderburn's little theorem
In the examples we came up with above, there're
examples of finite fields like
where is prime,
and example of infinite division ring like
the quaternions , but can you think
of any finite integral domain that is not
invertible, or any finite division ring that is
not commutative?
Potentially contrary to our intuition, the
Wedderburn's little theorem, which claims that
a finite domain must be a field, annihilates
the possibility of constructing a
non-invertible finite integral domain or a
non-commutative finite division ring.
The proof is initially done by representation
theory due to Wedderburn, but the version shown
in this text is due to Witt. Witt's proof is
done by discovering the numberic relationship
between the size of finite domain's unit group,
and the size of its center, and deriving
contradiction when the center is not trivially
the unit group. For me, the proof is truly a
delicate example of how to apply group theory
to solve ring theory problems.
Finite domains are division rings
For a finite domain , let
be the set of
non-zero elements in . By the domain's
property, we have
.
Given a non-zero element , consider
the map
defined by 's left
multiplication that ,
assume it is not injective that
,
then by there's
and , contradicting
to our previous assumption. So for every map
defined by non-zero 's left
multiplication is injective. And since the
image of contains elements,
every element in must be mapped and
is surjective, and thus bijective.
On the other hand, since every is a
bijective map on a finite set , and their
composition follows the associative rule of
the ring's multiplication, every can
actually be seen as a permutation
. Every permutation has
finite order, and if the order of is
, then , and
.
So and the finite domain
is a division ring.
By now, we've transform the original problem
into showing every finite division ring is a
field, that means, we need to show the unit
group of a finite division ring is abelian.
Finite division rings are vector spaces over their centers
Consider the unit group of a
finite division ring , assume
is the center of unit group
, define
as the center of domain , and .
It won't be hard to show the addition of
is closed and thus :
, and
.
The multiplication of is commutative
so is a field. And obviously there're
and
.
We need to show that is a vector space
over , which means all elements in
can be spanned from
where
,
, and
are deliberately selected basis, so that we
have .
Consider a algorithm described as below:
- Initialize that .
- If , the algorithm halts.
- Arbitrarily select , let
,
update and set
,
and back to step 2.
Obviously by the closed property of ,
for every . And if we
are able to show that
,
and
for every , then when the algorithm terminates
at , we have .
Assume there exists
,
where
are from , we have
and
.
However by
for every ,
there's , but such
is impossible to be selected from and
is a contradiction. So we have
.
Consider the and ,
if and
hold for
the cases up to by mathematical induction,
then we have
if every combination of
is
distinct. And by
,
the induction also holds for . So all
we need to do is to show every combination
in is distinct.
Assume there exists
where there's either or
, by moving terms we get
.
When , there're
and ,
violating our presumptions. When
,
is not zero and is
invertible, so we have
,
which implies and
is a contradiction. So every combination of
is distinct and ,
.
For the centralizer subgroup of
, it won't
be hard to show the addition of
is closed and
thus :
, and
.
Since the is
a division ring and
,
it's also a vector space over and
there's .
Finite division rings are fields
The class equation of can be
written as:
To understand the underlying numberic
relationship in this equation, we need to
understand some basic properties of
cyclotomic polynomials first.
We are familiar with the sum that
.
It won't be hard to see that for any
dividing , we can furtherly
factorize with a change of
variable that
.
However this also means another factorization
is possible that
.
Sometimes it's a dilemma picking up which
change of variable, like:
On the other hand, for , we can prove
divides iff divides
. Assume does not divide and
, we have
.
And since for
, we have
.
So when , divides
iff divides .
Define
,
the roots of cyclotomic equation
are called the -th root
of unity and in the form of
,
and there's
.
Obviously for any ,
we have ,
which is contained in
as a factor of .
An intuitive thinking is to partition
by their greatest common
divisor with , so that:
While for each
,
by dividing by in it we have
,
which is a function related to only
. So by defining
,
which is called the -th
cyclotomic polynomial, we have:
Which provides an alternative way of
viewing . Those roots of unity
that are contained in cyclotomic
polynomial are also called
the -th primitive roots of unity.
Next, let's show that all coefficients
in are integers, so that
.
This turns cyclotomic polynomials into
a powerful tool for factorizing
integrally.
First, we have ,
by transforming the Bezout's identity that
.
So when , for every -th
primitive root , there's
always another root
paired with it. Let
,
for every pair of
, we have
.
So in , the
coefficient of the highest term and
constant term are , and coefficients
in between are real-valued. Specially in
and
, the coefficients
of the highest term are also , the
coefficients in between are also
real-valued, but the constant terms
may be or .
Then, for the intermediate terms in
, although there're
instances like
and
, it's still
hard to see how these s
multiply together and yield integral
coefficients directly. So instead we
prove by mathematical induction: starting
from , assume
coefficients in
are integers, and see whether it implies
the coefficients in are
also integers.
Rewrite with :
Let ,
Recall that constant terms in cyclotomic
polynomials are either (in
) or (in
), and is the product
of cyclotomic polynomials, so we have
and:
Given that
,
starting from , assume
,
by mathematical induction we have
. By now we've
proved that coefficients of
are integers, so there's
.
Finally, consider each term's divisibility
by in the class equation.
is undoubtedly divisible by
. And since
contains a factor of ,
it's thereby divisible by .
Once we do the congruence
modulo- operation on each
sides of the class equation, we have:
When , .
When , for each pair of primitive
-th root
in , since
is real-valued, we have:
So in our problem, when , we
always have ,
and thus
,
which is a contradiction. The class
equation holds only when , which
implies , the
quotient ring is commutative, and thereby
a field.
Conclusion
In this text, we first introduce the concept
of rings, the zero divisors and units of
a ring, and the categorization of rings into
domains, division rings, integral domains and
fields, based on its zero divisors and units.
Then we discuss about the characteristic of a
ring, alongside with the constraints and
consequences of it.
Finally, we walk through Witt's proof of the
Wedderburn's little theorem: first we proved
there's no distinction between "no non-trivial
zero divisor" and "all elements are invertible"
in a finite ring; then we proved finite
division rings are vector space over its
center, deriving their size constraints;
finally by putting the sizes of unit group,
center and non-trivial orbits into the class
equation, we found it's impossible for the
equation to hold when there're non-trivial
orbits. By the theorem we know it's impossible
to construct non-invertible finite integral
domain or non-commutative finite division ring.
All these concepts and theorems are fundamental
in ring theory and will surely linger from time
to time, so understanding and mastering them
are mandatory.