Let R be a ring, and let I be a (two-sided) ideal. Considering just
the operation of addition, R is a group and I is a subgroup. In fact,
since R is an abelian group under addition, I is a
normal subgroup, and the quotient group is defined. Addition of cosets is defined by
adding coset representatives:
The zero coset is , and the additive inverse of a
coset is given by
.
However, R also comes with a multiplication, and it's natural to ask
whether you can turn into a ring
by multiplying coset representatives:
I need to check that that this operation is well-defined, and that the ring axioms are satisfied. In fact, everything works, and you'll see in the proof that it depends on the fact that I is an ideal. Specifically, it depends on the fact that I is closed under multiplication by elements of R.
By the way, I'll sometimes write " " and
sometimes "
"; they mean the same thing.
Theorem. If I is a two-sided ideal in a ring
R, then has the structure of a ring under coset addition and
multiplication.
Proof. Suppose that I is a two-sided ideal in
R. Let .
Coset addition is well-defined, because R is an abelian group and I a normal subgroup under addition. I proved that coset addition was well-defined when I constructed quotient groups.
I need to show that coset multiplication is well-defined:
As before, suppose that
Then
The next-to-last equality is derived as follows: , because I is an ideal; hence
. Note that this uses the
multiplication axiom for an ideal; in a sense, it explains why the
multiplication axiom requires that an ideal be closed under
multiplication by ring elements on the left and right.
Thus, coset multiplication is well-defined.
Verification of the ring axioms is easy but tedious: It reduces to the axioms for R.
For instance, suppose I want to verify associativity of
multiplication. Take . Then
(Notice how I used associativity of multiplication in R in the middle
of the proof.) The proofs of the other axioms are similar.
Definition. If R is a ring and I is a
two-sided ideal, the quotient ring of R mod I
is the group of cosets with the
operations of coset addition and coset multiplication.
Proposition. Let R be a ring, and let I be an ideal
(a) If R is a commutative ring, so is .
(b) If R has a multiplicative identity 1, then is a multiplicative identity for
. In this case, if
is a unit, then so
is
, and
.
Proof. (a) Let .
Since R is commutative,
Therefore, is commutative.
(b) Suppose R has a multiplicative identity 1. Let . Then
Therefore, is the identity of
.
If is a unit, then
Therefore, .
Example. ( A quotient ring of
the integers) The set of even integers is an ideal in
. Form the quotient ring
.
Construct the addition and multiplication tables for the quotient ring.
Here are some cosets:
But two cosets and
are the same exactly when a and b differ by an even
integer. Every even integer differs from 0 by an even integer. Every
odd integer differs from 1 by an even integer. So there are really
only two cosets (up to renaming):
and
.
Here are the addition and multiplication tables:
You can see that is isomorphic to
.
In general, is isomorphic to
. I've been using "
" informally to mean the set
with addition and multiplication mod n, and
taking for granted that the usual ring axioms hold. This example
gives a formal contruction of
as the quotient
ring
.
Example. is the ring
of polynomials with coefficients in
. Consider the
ideal
.
(a) How many elements are in the quotient ring ?
(b) Reduce the following product in to the form
:
(c) Find in
.
The ring is analogous to
. In the case of
, you do
computations mod n: To "simplify", you divide the result of
a computation by the modulus n and take the remainder. In
, the polynomial
acts like the "modulus". To do
computations in
, you divide the result of a computation by
and take the remainder.
(a) By the Division Algorithm, any can
be written as
This means that , where
. Then
Since there are 3 choices for a and 3 choices for b, there are 9
cosets.
(b) First, multiply the coset representatives:
Dividing by
, I get
Then
(c) To find multiplicative inverses in , you use the Extended Euclidean Algorithm. The same
idea works in quotient rings of polynomial rings.
Thus,
Example. (a) List the elements of the cosets
of in the ring
.
(b) Is the quotient ring an integral domain?
(a) If x is an element of a ring R, the ideal consists of all multiples of x by elements
of R. It is not necessarily the same as the additive subgroup
generated by x, which is
In this example, the additive subgroup generated by is
As usual, I get it by starting with the zero element and the generator
, then adding
until I get back to
.
This set is contained in the ideal ; I need to check whether it is the
same as the ideal.
If , then
Thus, an element of the ideal
consists of a pair
, where each component is even.
There are two even elements in
(namely 0 and
2) and 3 even elements in
(namely 0, 2,
and 4), so there are
such pairs. Thus, the ideal
has a maximum of 6 elements. Since
the additive subgroup above already has 6 elements, it must be the
same as the ideal.
I can list the elements of the cosets of the ideal as I would for subgroups.
(b) Note that
Hence, is not an integral domain.
Example. In the ring , consider the principal ideal
.
(a) List the elements of .
(b) List the elements of the cosets of .
(c) Is the quotient ring a field?
(a) Note that the additive subgroup generated by has only two elements. It's not the same as the ideal
generated by
, so I can't find the elements of the ideal
by taking additive multiples of
. I'll find the
elements of the ideal
by multiplying
by the elements of
, then throwing out duplicates. The computation is
routine, if a bit tedious.
Removing duplicates, I have
(b) Since the ideal has 4 elements and the ring has 20, there must be 5 cosets.
(c) Note that is the identity.
Since every nonzero coset has a multiplicative inverse, the quotient
ring is a field.
Copyright 2018 by Bruce Ikenaga