Page 152 - Advanced Linear Algebra

136 Advanced Linear Algebra

$~ ºº »»
=

~
The last coordinate of this sum is

Á ~

~

=
and so the difference # c $ has last coordinate and is thus in : ~ ºº »» .

Hence
= = r = »»
# ~ ²# c $³ b $  ºº »» b ºº »» ~ ºº =
as desired.
)
For part 2 , we leave it to the reader to review the proof and make the necessary
changes. The key fact is that : is isomorphic to an ideal of 9 , which is
:
principal. Hence, is generated by a single element of 4 .
The Hilbert Basis Theorem

Theorem 5.8 naturally leads us to ask which familiar rings are Noetherian. The
following famous theorem describes one very important case.

Theorem 5.9 Hilbert basis theorem) If a ring is Noetherian, then so is the
(
9
polynomial ring 9´%µ .
?
Proof. We wish to show that any ideal in 9´%µ is finitely generated. Let 3
?
denote the set of all leading coefficients of polynomials in , together with the
element of . Then is an ideal of .
3
9
9

To see this, observe that if 3 is the leading coefficient of ²%³  ? and if
 9, then either ~ or else is the leading coefficient of ²%³  . In
?

either case,  3 . Similarly, suppose that  3 is the leading coefficient of

²%³  ?. We may assume that deg ²%³ ~ and deg ²%³ ~ , with  . Then
²%³ ~ % c ²%³ is in , has leading coefficient and has the same degree as?

²%³. Hence, either c is or c is the leading coefficient of

²%³ c ²%³  ? . In either case c  3.
Since is an ideal of the Noetherian ring , it must be finitely generated, say
9
3
3 ~ º ÁÃÁ ». Since  3, there exist polynomials ²%³  with leading?

%
coefficient . By multiplying each ² % ³ by a suitable power of , we may
assume that
¸
deg ²%³ ~ ~ max deg ²%³¹

for all ~ Á Ã Á .