Page 164 - Advanced Linear Algebra

148 Advanced Linear Algebra

%~² b ³%~ %  4
Hence 4~ 4 .

)
For part 1 , since gcd ² Á Ã Á ³ ~ , there exist scalars for which

b Ä b ~
and so for any %4 ,

% ~ ² bÄb ³%  4

~
Moreover, since the ² 4³ “ and the 's are pairwise relatively prime, it

follows that the sum of the submodules 4 is direct, that is,
4~ 4 l Ä l 4~ 4 l Ä l 4

As to the annihilators, it is clear that º » ‹ ann ² 4³ . For the reverse

inclusion, if  ann ² 4³ , then  ann ²4³ and so “ , that is,

“ and so  º ». Thus ann ² 4³ ~ º ».

As to uniqueness, we claim that ~ Ä is an order of 4 . It is clear that

annihilates 4 and so “ . On the other hand, 5 contains an element of
"

order and so the sum # ~ " bÄb" has order , which implies that

. Hence,
“ and are associates.
Unique factorization in 9 now implies that ~ and, after a suitable
is primary of
reindexing, that ~ and and are associates. Hence, 5

order . For convenience, we can write 5 as 5 . Hence,

5 ‹ ¸#  4 “ # ~ ¹ ~ 4
But if
5 l Äl5 ~ 4 lÄl4

for all .
and 5‹ 4 for all , we must have 5~ 4

For part 3), if ~ and ¢ 4 š 5 , then the map ¢ 4 ¦ 5 defined by

² bÄb ³ ~ ² ³bÄb ² ³

is an isomorphism and so 4š 5 . Conversely, suppose that ¢ 4š 5 . Then

4 5 and have the same annihilators and therefore the same order
~ Ä

Hence, part 1) and part 2) imply that ~ and after a suitable reindexing,