392
8. MODULES
the ring elements
a
i
are nonzero and noninvertible, and
a
i
divides
a
j
for
i
j
;
Show that for
k
1
,
p
k
1
M=p
k
M
Š
.R=.p//
m
k
.p/
, where
m
k
.p/
is the number of
a
i
that are divisible by
p
k
. Conclude
that the numers
m
k
.p/
depend only on
M
and not on the choice
of the direct sum decomposition
M
D
A
1
˚
A
2
˚
˚
A
s
.
(c)
Show that the numbers
m
k
.p/
, as
p
and
k
vary, determine
s
and
also determine the ring elements
a
i
up to associates. Conclude
that the invariant factor decomposition is unique.
8.5.8.
Let
M
be a finitely generated torsion module over a PID
R
. Let
m
be a period of
M
with irreducible factorization
m
D
p
m
1
1
p
m
s
s
. Show
that for each
i
and for all
x
2
MOEp
i
Ł
,
p
m
i
i
x
D
0
.
8.6. Rational canonical form
In this section we apply the theory of finitely generated modules of a
principal ideal domain to study the structure of a linear transformation of
a finite dimensional vector space.
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 Fall '08
 EVERAGE
 Linear Algebra, Algebra, Vector Space, finite dimensional vector, dimensional vector space

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