Page 156 - Advanced Linear Algebra
P. 156
140 Advanced Linear Algebra
that is, the orders of 4 are precisely the least common multiples of the
orders of and .
)
(
Proof. We leave proof of part 1) for the reader. For part 2), suppose that
²4³ ~ Á ²(³ ~ Á ²)³ ~ Á ~ lcm ² Á ³
Then and ( ~ ¸ ¹ ) ~ imply that and and so . On the
¸ ¹
other hand, annihilates both and and therefore also 4 ~ ( l ) . Hence,
(
)
and so is an order of 4 .
Cyclic Modules
The simplest type of nonzero module is clearly a cyclic module. Despite their
simplicity, cyclic modules will play a very important role in our study of linear
operators on a finite-dimensional vector space and so we want to explore some
of their basic properties, including their composition and decomposition.
Theorem 6.3 Let be a principal ideal domain.
9
)
9
1 If ºº#»» is a cyclic -module with annihilator º » , then the multiplication
map defined by ~ # is an -epimorphism with kernel º . »
¢9 ¦ º º#» »
9
Hence the induced map
9
¢ ¦ º º # » »
º»
defined by
² b º »³ ~ #
is an isomorphism. In other words, cyclic -modules are isomorphic to
9
quotient modules of the base ring .
9
)
2 Any submodule of a cyclic -module is cyclic.
9
)
3 If ºº#»» is a cyclic submodule of 4 of order , then for 9 ,
²ºº #»»³ ~
gcd ²Á ³
Also,
ºº #»» ~ ºº#»» ¯ ² ²#³Á ³ ~ ¯ ² #³ ~ ²#³
)
)
Proof. We leave proof of part 1 as an exercise. For part 2 , let : ºº#»» . Then
0~ ¸ 9 # :¹ is an ideal of 9 and so 0~ º » for some 9. Thus,
: ~0# ~9 # ~ºº #»»
)
For part 3 , we have ² #³ ~ if and only if ² ³# ~ , that is, if and only if
, which is equivalent to
