Page 153 - Advanced Linear Algebra

Modules II: Free and Noetherian Modules 137

Now for ~ Á Ã Á c let 3 be the set of all leading coefficients of
polynomials in of degree , together with the element of . A similar
?

9

argument shows that 3 is an ideal of and so 3 9 is also finitely generated.
²%³¹ in ? whose
Hence, we can find polynomials 7 ~ ¸ ²%³Á à Á Á

Á
.
leading coefficients constitute a generating set for 3
Consider now the finite set
c
7 ~ 8 7 9 r ¸ ²%³Á à Á ²%³¹

~
7
If @ is the ideal generated by , then @ ‹ ? . An induction argument can be
used to show that @ ? ~ . If ² ³  ? % has degree , then it is a linear

combination of the elements of 7 ( ) .
which are constants and is thus in @

Assume that any polynomial in of degree less than is in and let ² @ % ³  ?
?
have degree .

If  , then some linear combination ²%³ over of the polynomials in 7
9
has the same leading coefficient as ²%³ and if ‚ , then some linear
combination ²%³ of the polynomials
B c ²%³Á à Á % c ²%³ ‹ @
C%

has the same leading coefficient as ²%³ . In either case, there is a polynomial
²%³  @ ²%³. Since ²%³ c ²%³  ? that has the same leading coefficient as
has degree strictly smaller than that of ²%³ the induction hypothesis implies that
²%³ c ²%³  @
and so
²%³ ~ ´ ²%³ c ²%³µ b ²%³  @
?
This completes the induction and shows that ~ @ is finitely generated.
Exercises
1. If 4 is a free -module and ¢ 4 ¦ 5 is an epimorphism, then must 5
9
also be free?
?
2. Let be an ideal of . Prove that if ° 9 ? is a free -module, then is the
?
9
9
zero ideal.
3. Prove that the union of an ascending chain of submodules is a submodule.
:
4. Let be a submodule of an 9 -module 4 . Show that if 4 is finitely
generated, so is the quotient module 4°: .
5. Let be a submodule of an -module. Show that if both and 4 ° : are
9
:
:
finitely generated, then so is 4 .
6. Show that an -module 4 satisfies the ACC for submodules if and only if
9
the following condition holds. Every nonempty collection of submodules
I