Sections 5.1 through 5.4

- Commutative rings, in general
- Ideals and factor rings
- Integral domains

Sections 5.5 through 5.8

- Localization in integral domains
- Noncommutative examples
- Isomorphism theorems

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** 5.1.1. Definition **
Let R be a set on which two binary operations are
defined, called addition and multiplication, and denoted by + and
**·**.
Then R is called a
**commutative ring**
with respect to these operations if the following properties hold:

**(i)**
*Closure:*
If a,b R,
then the sum a+b and the product
a**·**b
are uniquely defined and belong to R.

**(ii)**
*Associative laws:*
For all a,b,c R,

a+(b+c) = (a+b)+c and
a**·**(b**·**c)
= (a**·**b)**·**c.

a+b = b+a and
a**·**b
= b**·**a.

a**·**(b+c)
= a**·**b
+ a**·**c and
(a+b)**·**c
= a**·**c
+ b**·**c.

a+0 = a and 0+a = a.

a+x = 0 and x+a = 0

have a solution x R, called theThe commutative ring R is called a

a**·**1 = a and
1**·**a = a.

As with groups, we will use juxtaposition to indicate multiplication,
so that we will write ab instead of
a**·**b.

**Example** 5.1.1.
(**Z**_{n})
The rings **Z**_{n}
form a class of commutative rings
that is a good source of examples and counterexamples.

**5.1.2. Definition**
Let S be a commutative ring.
A nonempty subset R of S is called a
**subring**
of S if it is a commutative ring under the addition and multiplication of S.

**5.1.3. Proposition**
Let S be a commutative ring, and let R be a nonempty subset of S.
Then R is a subring of S if and only if

**(i)**R is closed under addition and multiplication; and**(ii)**if a R, then -a R.

**5.1.5. Proposition**
Let R be a commutative ring with identity.
Then the set R^{×}
of units of R is an abelian group under the multiplication of R.

An element e of a commutative ring R is said to be
**idempotent**
if e^{2} = e.
An element a is said to be
**nilpotent**
if there exists a positive integer n with
a^{n} = 0.

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

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

for all a,b R.

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
RS.
An isomorphism from the commutative ring R onto itself is called an
**automorphism**
of R.

**5.2.2. Proposition**

**(a)**The inverse of a ring isomorphism is a ring isomorphism.**(b)**The composition of two ring isomorphisms is a ring isomorphism.

**(a)**(0) = 0;**(b)**(-a) = -(a) for all a in R;**(c)**if R has an identity 1, then (1) is idempotent;**(d)**(R) is a subring of S.

{ a R | (a) = 0 }

is called the
**5.2.5. Proposition**
Let :R->S be a ring homomorphism.

**(a)**If a,b ker() and r R, then a+b, a-b, and ra belong to ker().**(b)**The homomorphism is an isomorphism if and only if ker() = {0} and (R) = S.

(r) = (r) for all r R and (x) = s, defined by

(a_{0} +
a_{1}x
+ ... +
a_{m}x^{m}) =
(a_{0}) +
(a_{1})s
+ ... +
(a_{m})s^{m}.

**5.2.7. Proposition**
Let R and S be commutative rings.
The set of ordered pairs (r,s) such
that r R and
s S
is a commutative ring under componentwise addition and multiplication.

**5.2.8. Definition**
Let R and S be commutative rings.
The set of ordered pairs (r,s) such
that r R and
s S is called the
**direct sum**
of R and S.

**Example** 5.2.10.
The ring **Z**_{n}
is isomorphic to the direct sum of the rings
**Z**_{k}
that arise in the prime factorization of n.
This describes the structure of
**Z**_{n}
in terms of simpler rings,
and is the first example of what is usually called a
``structure theorem.''
This structure theorem can be used to
determine the invertible, idempotent, and nilpotent elements of
**Z**_{n}
and provides an easy proof of our earlier formula
for the Euler phi-function in terms of the prime factors of n.

**5.2.9. Definition**
Let R be a commutative ring with identity.
The smallest positive integer n such that (n)(1) = 0
is called the **characteristic** of R, denoted by char(R).
If no such positive integer exists,
then R is said to have **characteristic zero**.

**(i)**a ± b I for all a,b I, and**(ii)**ra I, for all a I and r R.

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

The ideal I is said to be a
**maximal ideal**
of R if for all ideals J of R such that
I J
R,
either J = I or J = R.

For an ideal I of a commutative ring R, the set
{ a+I | aR } of
cosets
of I in R (under addition) is denoted by R/I. By
Theorem 3.8.4,
the set forms a group under addition.
The next theorem justifies calling R/I the
**factor ring**
of R modulo I.

**5.3.6. Theorem**
If I is an ideal of the commutative ring R,
then R/I is a commutative ring, under the operations

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

for all a,b R.
**5.3.7. Proposition**
Let I be an ideal of the commutative 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 the ideals of R that contain I.

The ring of integers **Z**
is the most fundamental example of an integral domain.
The ring of all polynomials with real coefficients is also an integral domain,
but the larger ring of all real valued functions is not an integral domain.

The cancellation law for multiplication holds in R if and only if R has no nonzero divisors of zero. One way in which the cancellation law holds in R is if nonzero elements have inverses in a larger ring; the next two results characterize integral domains as subrings of fields (that contain the identity 1).

**5.1.7. Theorem**
Let F be a field with identity 1.
Any subring of F that contains 1 is an integral domain.

**5.4.4. Theorem**
Let D be an integral domain.
Then there exists a field F that contains a subring isomorphic to D.

**5.1.8. Theorem**
Any finite integral domain must be a field.

**5.2.10. Proposition**
An integral domain has characteristic 0 or p, for some prime number p.

**5.3.9. Proposition**
Let I be a proper ideal of the commutative ring R with identity.

**(a)**The factor ring R/I is a field if and only if I is a maximal ideal of R.**(b)**The factor ring R/I is a integral domain if and only if I is a prime ideal of R.**(c)**If I is maximal, then it is a prime ideal.

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

is called theAn integral domain in which every ideal is a principal ideal is called a

**Example** 5.3.1.
(**Z** is a principal ideal domain)
Theorem 1.1.4
shows that the ring of integers
**Z** is a principal ideal domain.
Moreover, given any nonzero ideal I of **Z**,
the smallest positive integer in I is a generator for the ideal.

**5.3.10. Theorem**
Every nonzero prime ideal of a principal ideal domain is maximal.

**Example** 5.3.7. (Ideals of F[x])
Let F be any field. Then F[x] is a principal ideal domain, since by
Theorem 4.2.2
the ideals of F[x] have the form I = <f(x)>,
where f(x) is the unique monic polynomial of minimal degree in the ideal.
The ideal I is prime (and hence maximal) if and only if f(x) is irreducible.
If p(x) is irreducible, then the factor ring
F[x]/<p(x)> is a field.

** Example ** 5.3.8. (Evaluation mapping)
Let F be a subfield of E, and for any element
u E define the evaluation mapping
_{u}:F[x]->E by
_{u}(g(x)) = g(u),
for all g(x) F[x]. Since
_{u}(F[x])
is a subring of E that contains 1, it is an integral domain,
and so the kernel of
_{u}
is a prime ideal.
Thus if the kernel is nonzero, then it is a maximal ideal, so
F[x]/ker(_{u})
is a field, and the image of
_{u}
is a subfield of E.