Page 39 - Advanced Linear Algebra
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Preliminaries 23
Definition An ideal in a ring is a maximal ideal if £ ? 9 and if whenever
?
9
@ ? is an ideal satisfying @ 9 @ ~ ? or @ , then either ~ 9 .
Here is one reason why maximal ideals are important.
Theorem 0.22 Let be a commutative ring with identity. Then the quotient
9
ring 9° ? is a field if and only if is a maximal ideal.
?
9
?
Proof. First, note that for any ideal of , the ideals of ° 9 ? are precisely the
°
quotients @? where is an ideal for which ? @ @ 9 . It is clear that @?
°
is an ideal of 9° ? . Conversely, if A Z is an ideal of 9° ? , then let
A ? ~¸ 9 b A Z ¹
A
It is easy to see that is an ideal of for which ? A 9 .
9
Next, observe that a commutative ring with identity is a field if and only if :
:
has no nonzero proper ideals. For if is a field and is an ideal of : ?
:
containing a nonzero element , then ~ ? c and so ? ~: . Conversely,
:
if has no nonzero proper ideals and £ : , then the ideal º » must be :
and so there is an : for which ~ . Hence, is a field.
:
Putting these two facts together proves the theorem.
The following result says that maximal ideals always exist.
Theorem 0.23 Any nonzero commutative ring 9 with identity contains a
maximal ideal.
Proof. Since is not the zero ring, the ideal ¸ ¹ is a proper ideal of . Hence,
9
9
the set of all proper ideals of is nonempty. If
9
I
9 ? ~¸ 0¹
9
?
is a chain of proper ideals in , then the union @ ~ 0 is also an ideal.
Furthermore, if @ is not proper, then @ ~9 and so ? , for some 0 ,
?
which implies that ~9 is not proper. Hence, @ I . Thus, any chain in I
has an upper bound in and so Zorn's lemma implies that has a maximal
I
I
9
element. This shows that has a maximal ideal.
Integral Domains
Definition Let be a ring. A nonzero element r 9 is called a zero divisor if
9
there exists a nonzero 9 for which ~ . A commutative ring with
9
identity is called an integral domain if it contains no zero divisors.
has zero divisors and
Example 0.14 If is not a prime number, then the ring {
so is not an integral domain. To see this, observe that if is not prime, then
~ in , where Á . But in , we have
{
{
