Page 134 - Advanced Linear Algebra
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118 Advanced Linear Algebra
Theorem 4.7 Let be a submodule of an -module 4 . The binary relation
9
:
"#¯" c #:
is an equivalence relation on 4 , whose equivalence classes are the cosets
# b : ~¸# b :¹
of in 4 . The set 4 ° : of all cosets of in 4 , called the quotient module of
:
:
4 : modulo 9 , is an -module under the well-defined operations
²" b:³b²#b:³ ~ ²"b#³b:
²"b:³ ~ "b:
The zero element in 4°: is the coset b : ~ : .
One question that immediately comes to mind is whether a quotient module of a
free module must be free. As the next example shows, the answer is no.
{
Example 4.6 As a module over itself, is free on the set ¸ ¹ . For any ,
the set { { ~¸' ' ¹ is a free cyclic submodule of , but the quotient -
{
{
module {{° is isomorphic to { via the map
{²" b ³ ~ " mod
{
{
{
{
and since is not free as a -module, neither is ° .
The Correspondence and Isomorphism Theorems
The correspondence and isomorphism theorems for vector spaces have analogs
for modules.
Theorem 4.8 The correspondence theorem) Let be a submodule of 4 .
(
:
Then the function that assigns to each intermediate submodule : ; 4 the
(
quotient submodule ;°: of 4°: is an order-preserving with respect to set
inclusion one-to-one correspondence between submodules of 4 containing :
)
and all submodules of 4°: .
(
Theorem 4.9 The first isomorphism theorem ) Let ¢4 ¦ 5 be an 9 -
Z
homomorphism. Then the map ¢4°ker ² ³ ¦ 5 defined by
²# b ker
Z ² ³³ ~ #
is an -embedding and so
9
4
ker ²³ ² im ³
