Page 128 - Advanced Linear Algebra
P. 128
112 Advanced Linear Algebra
Theorem 4.1 A nonempty subset of an -module 4 is a submodule if and
9
:
only if it is closed under the taking of linear combinations, that is,
Á 9Á "Á # : ¬ " b # :
Theorem 4.2 If and are submodules of 4 , then q : ; and b : ; are also
:
;
submodules of 4 .
We have remarked that a commutative ring with identity is a module over
9
itself. As we will see, this type of module provides some good examples of non-
vector-space-like behavior.
When we think of a ring as an -module rather than as a ring, multiplication
9
9
is treated as scalar multiplication. This has some important implications. In
particular, if is a submodule of , then it is closed under scalar multiplication,
:
9
which means that it is closed under multiplication by all elements of the ring .
9
In other words, is an ideal of the ring . Conversely, if is an ideal of the
:
?
9
ring , then is also a submodule of the module . Hence, the submodules of
?
9
9
the -module are precisely the ideals of the ring .
9
9
9
Spanning Sets
The concept of spanning set carries over to modules as well.
(
Definition The submodule spanned or generated) by a subset of a module
:
: is the set of all linear combinations
4 of elements of :
ºº:»» ~ ¸ # bÄb # 9Á # :Á ¹
A subset : 4 is said to span 4 or generate 4 if 4 ~ ºº:»» .
We use a double angle bracket notation for the submodule generated by a set
-
because when we study the -vector space/ ´ - % µ -module = , we will need to
make a distinction between the subspace º#» ~ -# generated by # = and the
#
submodule ºº#»» ~ -´%µ# generated by .
One very important point to note is that if a nontrivial linear combination of the
9
elements #Á Ã Á # in an -module 4 is ,
# bÄb # ~
where not all of the coefficients are , then we cannot conclude, as we could in
a vector space, that one of the elements is a linear combination of the others.
#
After all, this involves dividing by one of the coefficients, which may not be
{
{
possible in a ring. For instance, for the -module d { we have
² Á ³ c ² Á ³ ~ ² Á ³
but neither ² Á ³ nor ² Á ³ is an integer multiple of the other.
