Page 170 - Advanced Linear Algebra

154 Advanced Linear Algebra

)
1 If 4 has order

~ Ä

where the 's are distinct nonassociate primes in 9 , then 4 can be
(
uniquely decomposed up to the order of the summands into the direct sum
)

4~ 4 l Ä l 4
where

4 ~ 4 ~ ¸#  4 “ # ~ ¹

is a primary submodule with annihilator º » . Finally, each primary

can be written as a direct sum of cyclic submodules, so that
submodule 4
’•••••••••“•••••••••” ’•••••••••“•••••••••”
4 ~ ´ ºº# »» lÄlºº# Á »» µlÄl´ºº# Á »»lÄlºº# Á »»µ
Á

4 4
Á
where ann²ºº# »»³ ~ º » and the terms in each cyclic decomposition can
Á

be arranged so that, for each ,

²º
²
ann º# »»³ ‹ Ä ‹ ann ºº# Á »»³
Á
or, equivalently,
~ ‚ Á ‚ Á Ä ‚ Á
2 As for uniqueness, suppose that
)
»»µ
’•••••••••“•••••••••”
»»lÄlºº"
Á

Á

4 ~ ´ºº" »» lÄlºº" Á »»µlÄl´ºº" ’••••••••••“••••••••••”
Á

5 5
is also a primary cyclic decomposition of 4 . Then,
a The number of summands is the same in both decompositions; in fact,
)
~ for all ".
~ and after possible reindexing, " "
)
b The primary submodules are the same; that is, after possible
reindexing, — and 5 ~ 4

c For each primary submodule pair 5~ 4 , the cyclic submodules
)
have the same annihilator chains; that is, after possible reindexing,
ann²ºº" »»³ ~ ann²ºº# »»³
Á
Á
for all Á .
In summary, the primary submodules and annihilator chains are uniquely
determined by the module 4 .
3 Two -modules 4 and are isomorphic if and only if they have the same
)
5
9
annihilator chains.