Page 136 - Advanced Linear Algebra
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120 Advanced Linear Algebra
2 )(Independence of the family ) For each 0 ,
p s
: q : ~ ¸ ¹
q £ t
In this case, each is called a direct summand of 4 . If < ~ ¸ : Á Ã Á : ¹ is
:
a finite family, the direct sum is often written
4~ : l Ä l :
:
;
Finally, if 4~ : l ; , then is said to be complemented and is called a
complement of in 4 .
:
As with vector spaces, we have the following useful characterization of direct
sums.
9
Theorem 4.12 Let < ~¸: 0¹ be a family of distinct submodules of an -
module 4 . The following are equivalent:
1 )(Independence of the family ) For each 0 ,
p s
: q : ~ ¸ ¹
q £ t
2 )(Uniqueness of expression for ) The zero element cannot be written as
a sum of nonzero elements from distinct submodules in .
<
3 )(Uniqueness of expression ) Every nonzero #4 has a unique, except for
order of terms, expression as a sum
# ~ bÄb
of nonzero elements from distinct submodules in .
<
Hence, a sum
4~ :
0
)
)–
is direct if and only if any one of 1 3 holds.
In the case of vector spaces, every subspace is a direct summand, that is, every
subspace has a complement. However, as the next example shows, this is not
true for modules.
Example 4.7 The set of integers is a -module. Since the submodules of {
{
{
are precisely the ideals of the ring and since is a principal ideal domain, the
{
{
submodules of are the sets
{
ºº »»~ ~¸' ' ¹
{
{
