Page 141 - Advanced Linear Algebra

Modules I: Basic Properties 125

)
5 There exist free modules with submodules that are not free.
)
6 There exist free modules with linearly independent sets that are not
contained in a basis and spanning sets that do not contain a basis.
Recall also that a module over a noncommutative ring may have bases of
different sizes. However, all bases for a free module over a commutative ring
with identity have the same size, as we will prove in the next chapter.
Exercises
1. Give the details to show that any commutative ring with identity is a
module over itself.
2. Let : ~ ¸# ÁÃÁ# ¹ be a subset of a module 4 . Prove that 5 ~ ºº:»» is

the smallest submodule of 4 containing . First you will need to formulate
:
precisely what it means to be the smallest submodule of 4 containing .
:
3. Let 4 be an -module and let be an ideal in . Let 4 0 be the set of all
9
0
9
finite sums of the form

# bÄb #

where  0 and #  4 . Is 04 a submodule of 4 ?

(
4. Show that if and are submodules of 4 , then with respect to set
:
;
inclusion)
:q ; ~ glb ¸:Á ;¹ and :b ; ~ lub ¸:Á ;¹
5. Let :‹ :‹ Ä be an ascending sequence of submodules of an 9 -

module 4 . Prove that the union : is a submodule of 4 .
6. Give an example of a module 4 that has a finite basis but with the property
that not every spanning set in 4 contains a basis and not every linearly
independent set in 4 is contained in a basis.
7. Show that, just as in the case of vector spaces, an -homomorphism can be
9
defined by assigning arbitrary values on the elements of a basis and
extending by linearity.
9
8
8. Let  ² hom 9 4 Á 5 ³ be an -isomorphism. If is a basis for 4 , prove
5
that 8 ~ ¸ “  8¹ is a basis for .
9
9. Let 4 be an -module and let  hom 9 ²4Á 4³ be an -endomorphism.

9
If is idempotent , that is, if ~ , show that

4~ ker ² ³ l im ² ³

Does the converse hold?
10. Consider the ring 9~ -´%Á &µ of polynomials in two variables. Show that
the set 4 consisting of all polynomials in that have zero constant term is
9
an -module. Show that 4 is not a free -module.
9
9
11. Prove that if 9 is an integral domain, then all -modules 4 9 have the
is linearly independent over , then so is
following property: If #Á Ã Á # 9
for any nonzero  9.
# ÁÃÁ #