Page 157 - Advanced Linear Algebra

Modules Over a Principal Ideal Domain 141

• d
gcd ²Á
³
Thus,  ann ² #³ if and only if  º » and so ann ² #³ ~º » . For the second

statement, if ²Á ³ ~ then there exist Á  9 for which b ~ and so

# ~ ² b ³# ~ #  ºº #»» ‹ ºº#»»
and so ºº #»» ~ ºº#»» . Of course, if ºº #»» ~ ºº#»» then ² #³ ~ . Finally, if

² #³ ~ , then

~ ² #³~
gcd ²Á
³
and so ²Á ³ ~ .

The Decomposition of Cyclic Modules
The following theorem shows how cyclic modules can be composed and
decomposed.
Theorem 6.4 Let 4 be an -module.
9
1 )(Composing cyclic modules ) If "Á à Á "  4 have relatively prime

orders, then
²" b Ä b " ³ ~ ²" ³Ä ²" ³

and
ºº" »» l Ä l ºº" »» ~ ºº" b Ä b " »»

Consequently, if
4~ ( b Ä b (
have relatively prime orders, then the sum is
where the submodules (
direct.
2 )(Decomposing cyclic modules ) If ²#³ ~ Ä where the 's are
pairwise relatively prime, then has the form
#
# ~ " bÄb"
where ²" ³ ~ and so

ºº#»» ~ ºº" b Ä b " »» ~ ºº" »» l Ä l ºº" »»

Proof. For part 1), let ~ , ³ ²" • Ä and # • " bÄb" . Then

#

since annihilates , the order of divides . If ² # ³ is a proper divisor of ,
#

then for some index , there is a prime “ for which ° annihilates . But
#
° annihilates each for £ . Thus,
"