Sections 5.5 through 5.8

- Localization in integral domains
- Noncommutative examples
- Isomorphism theorems

Forward | Back | Table of Contents | About this document

It is important to note that we will always assume that our rings have a multiplicative identity element. Furthermore, in what represents a significant change from the previous sections, we will assume that a subring always has the same multiplicative identity as the ring itself, and that ring homomorphisms preserve the multiplicative identity element.

D

is an integral domain with D D

** 5.8.10. Definition. **
Let D be an integral domain with quotient field Q(D),
and let P be a prime ideal of D. Then

D_{P}
= { ab^{-1}
Q(D) |
b P }

is called the
** localization** of D at P.

If I is any ideal of D, then

I_{P}
= { ab^{-1}
Q(D) |
a I and
b P }

is called the
** extension ** of I to
D_{P}.

** 5.8.11. Theorem. **
Let D be an integral domain with quotient field Q(D),
and let P be a prime ideal of D.

**(a)**
If J is any proper ideal of
D_{P},
then there exists an ideal I of D with
I P
such that J = I_{P}.
If J is a prime ideal, then so is I.

**(b)**
If J is any prime ideal of D with
J P,
then J_{P}
is a prime ideal of
D_{P} with
D J_{P} = J.

**(c)**
There is a one-to-one correspondence between prime ideals of
D_{P}
and prime ideals of D that are contained in P.
Furthermore,
P_{P}
is the unique maximal ideal of
D_{P}.

** Example. ** 5.5.1.
(Endomorphisms of abelian groups)

Let A be an abelian group,
with its operation denoted by +.
Let R be the set of all endomorphisms of A.
That is, R is the set of all group homomorphisms f:A->A.
We can define addition and multiplication of elements of R as follows:
if f,g R, then

(f+g) (x) = f(x) + g(x) and
(f ^{.} g) (x) = f(g(x))

for all x A.

We need to repeat Definition 5.1.1, dropping the assumption that multiplication is commutative. We will, however, assume that all rings have a multiplicative identity element. It can then be shown that the set of all endomorphisms of an abelian group A, denoted by End(A), forms a ring.

** Definition **
A ** ring ** R is a set with two binary operations,
denoted by addition and multiplication,
such that the following properties hold:

**(i)**
For all a,b,c R,
a+(b+c) = (a+b)+c and a(bc) = (ab)c.

**(ii)**
For all a,b R,
a+b = b+a.

**(iii)**
For all a,b,c R,
a(b+c) = ab+ac and (a+b)c = ac+bc.

**(iv)**
The set R contains an
** additive identity element**,
denoted by 0, and a
** multiplictive identity element**,
denoted by 1,
such that a+0 = a, 1a = a, and a1=a,
for all a R.

**(v)**
For each a R,
the equation a+x = 0 has a solution x = -a in R, the
** additive inverse ** of a.

** Example. ** 5.5.2.
(Polynomial Rings)

Let R be any ring.
We let R[x] denote the set of infinite tuples

(a_{0},a_{1},a_{2},...)

such that
a_{i} R
for all i, and
a_{i} 0
for only finitely many terms a_{i}.
Two sequences are equal if and only if all corresponding terms are equal.
We introduce addition and multiplication as follows:

(a_{0},a_{1},a_{2},...) +
(b_{0},b_{1},b_{2},...) =
(a_{0}+b_{0},a_{1}+b_{1},a_{2}+b_{2},...)

(a_{0},a_{1},a_{2},...)
^{.}
(b_{0},b_{1},b_{2},...)
= (c_{0},c_{1},c_{2},...),

where
c_{n}
=
a_{i}
b_{n-i}.

With these operations it can be shown that R[x] is a ring.

We can identify a R with
(a,0,0,...) R[x],
and so if R has an identity 1,
then (1,0,0,...) is an identity for R[x].
If we let x=(0,1,0,...), then the elements of R[x]
can be expressed in the form

a_{0} +
a_{1} x + . . . +
a_{m-1}
x^{m-1} +
a_{m}
x^{m},

allowing us to use our previous notation for the
** ring of polynomials over R in the indeterminate x**.
Note that although the elements of R need not commute with each other,
they do commute with the indeterminate x.

If n is the largest nonnegative integer such that
a_{n} 0,
then we say that the polynomial has
** degree n**,
and a_{n} is called the
** leading coefficient **
of the polynomial.

** Example. ** 5.5.3.
(Differential operator rings)

Consider the homogeneous linear differential equation

a_{n}(x)
D^{n} y + . . . +
a_{1}(x) D y +
a_{0}(x) y = 0,

where the solution y(x) is a polynomial with complex coefficients,
and the terms a_{i}(x) also belong to
* C*[x].
The equation can be written in compact form as L(y)=0,
where L is the differential operator

a

with D = d/dx. Thus the differential operator can be thought of as a polynomial in the two indeterminates x and D, but in this case the indeterminates do not commute, since

D(x y(x)) = y(x) + x D(y(x)),

yielding the identity

Dx=1+xD.

Repeated use of this identity makes it possible to write the composition of two differential operators in the standard form

a

and we denote the resulting ring by

** Example. ** 5.5.4.
(Group algebras)

Let K be a field, and let G be a finite group of order n,
with elements 1=g_{1},
g_{2}, . . . ,
g_{n}. The
** group algebra **
KG is defined to be the n-dimensional vector space over K
with the elements of G as a basis.
Vector addition is used as the addition in the ring.
Elements of KG can be described as sums of the form

c_{i}
g_{i}

and multiplication is defined as for polynomials,
where the product
g_{i} g_{j}
is given by the product in G.

** Example. ** 5.5.5.
(Matrix rings)

Let R be a ring.
We let M_{n}(R) denote the set of all
n×n matrices with entries in R.

For
[a_{ij}]
and
[b_{ij}]
in M_{n}(R),
we use componentwise addition

[a_{ij}] +
[b_{ij}]
=
[a_{ij}+b_{ij}]

and the multiplication is given by

[a_{jk}]
[b_{jk}]
=
[c_{jk}],

where
[c_{ij}]
is the matrix whose j,k-entry is

c_{jk}
=
a_{ji}
b_{ik}.

** 5.5.3. Definition. **
Let R be a ring with identity 1, and let
a R.
If ab=0 for some nonzero b R,
then a is called a
** left zero divisor**.
Similarly, if ba=0 for some nonzero
b R,
then a is called a
** right zero divisor**.
If a is neither a left zero divisor nor a right zero divisor,
then a is called a
** regular** element.

The element a R is said to be
** invertible**
if there exists an element
b R
such that ab=1 and ba=1.
The element a is also called a
** unit**
of R, and its multiplicative inverse is usually denoted by
a^{-1}.

The set of all units of R is denoted by
R^{×}.

** 5.5.4. Proposition. **
Let R be a ring.
Then the set R^{×} of units of R
is a group under the multiplication of R.

** 5.5.5. Definition. **
A ring R in which each nonzero element is a unit is called a
** division ring**
or ** skew field**.

** Example. ** 5.5.8.
(The quaternions)

The following subset of
M_{2}(* C*)
is called the set of

Q = a + b + c + d

See Example 3.3.7 for the group of quaternion units.

The subset I is called a

The subset I is called a

For any ring R, it is clear that the set {0} is
an ideal, which we will refer to as the
** trivial**
ideal.
Another ideal of R is the ring R itself.

** 5.6.2. Definition. **
Let R be a ring, and let a R.
The left ideal

Ra = { x R | x = ra
for some r R }

is called the
** principal left ideal**
generated by a.

** 5.6.3. Proposition. **
Let R be a ring, and let I,J be left ideals of R.
The following subsets of R are left ideals.

**(a)**
I J;

**(b)**
I + J =
{ x R | x = a + b for some
a I,
b J };

**(c)**
IJ = {
a_{i}
b_{i} |
a_{i}
I,
b_{i}
J,
n
**Z**^{+} }.

** Example. ** 5.6.1.
(Ideals of M_{n}(R))

Let R be a ring, and let
M_{n}(R)
be the ring of matrices over R.
If I is an ideal of R, then the set
M_{n}(I)
of all matrices with entries in I is an ideal of S.
Conversely, every ideal of S is of this type.

** 5.6.4. Theorem. **
If I is an ideal of the ring R,
then R/I is a ring.

** 5.6.5. Definition. **
Let I be an ideal of the ring R.
With the following addition and multiplication for all
a,b R,
the set of cosets

{ a+I | a R } is denoted by R/I,
and is called the
** factor ring**
of R modulo I.

(a+I) + (b+I) = (a+b) + I and
(a+I)(b+I) = ab + I

Let I be a proper ideal of the ring R.
Then I is said to be a
** completely prime ideal**
of R if for all
a,b R it is true that
ab I implies
a I or
b I.

As in the commutative case,
a ring is called a
** domain**
if (0) is a completely prime ideal.
An element c of R is said to be
** regular**
if xc = 0 or cx = 0 implies x = 0, for all
x R.
Thus a ring is a domain if and only if
every nonzero element is regular.

** 5.7.1. Definition. **
Let R and S be rings. A function
:R->S is called a
** ring homomorphism**
if

**(i)**
(a+b) =
(a) +
(b), for all
a,b R,

**(ii)**
(ab) =
(a)
(b), for all
a,b R, and

**(iii)**
(1) = 1.

A ring homomorphism that is one-to-one and onto is called an
** isomorphism**.
If there is an isomorphism from R onto S, we say that R is
** isomorphic**
to S, and write R S.
An isomorphism from the ring R onto itself is called an
** automorphism**
of R.

** 5.7.2. Proposition. **
Any ring R is isomorphic to a subring of an endomorphism ring End(A),
for some abelian group A.

** 5.7.4. Definition. **
Let :R->S be a ring homomorphism.
The set

{ a R |
(a) = 0 }

is called the
** kernel**
of ,
denoted by ker().

** 5.7.5. Proposition. **
Let :R->S be a ring homomorphism.

**(a)**
If a,b
ker() and
r R, then
a+b, a-b, ra, and ar belong to ker().

**(b)**
The homomorphism
is an isomorphism if and only if
ker()={0} and
(R)=S.

** 5.7.6. Proposition. **
Let R and S be rings, let
:R->S be a ring homomorphism,
and let

:{
x_{1},
x_{2}, . . . ,
x_{n}}->Z(S)
be any mapping into the center of S.
Then there exists a unique ring homomorphism

:
R [ x_{1},
x_{2}, . . . ,
x_{n}] -> S
such that

(r) =
(r) for all
r R and

(x_{i}) =
(x_{i}),
for i=1,2,...,n.

** Example. ** 5.7.2.
Let G and H be finite groups, and let K be a field. If
:G->H is a group homomorphism,
we can extend the mapping
to a ring homomorphism
:KG->KH as follows:

(
_{x G}
c_{x} x ) =
_{x G}
c_{x} (x).

** 5.7.7. Theorem.
[Fundamental Homomorphism Theorem for Rings]
**
Let
:R->S be a ring homomorphism.
Then (R) is a subring of S,
R/ker() is a ring, and
(R)R/ker().

** 5.7.8. Proposition. **
Let I be an ideal of the ring R.

**(a)**
The natural projection mapping
:R->R/I defined
by (a)=a+I for all
a R is
a ring homomorphism, and
ker() = I.

**(b)**
There is a one-to-one correspondence between the ideals
of R/I and ideals of R that contain I.

**(c)**
If K is an ideal of R with
I K R,
then (R/I)/(K/I)R/K.

** 5.7.9. Theorem.
[Chinese Remainder Theorem]
**
Let R be a ring, and let
I_{1},
I_{2} be ideals of R such
I_{1}+I_{2}=R.
Then

( R / I_{1})
( R /
I_{2})
R / (I_{1}
I_{2}).