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8.5. FINITELY GENERATED MODULES OVER A PID, PART II.
385
as an
R=.p/
–vector space. Applying the same considerations to the other
direct sum decomposition, we obtain that the number of
b
i
divisible by
p
is also equal to dim
R=.p/
.M=pM/
.
If
p
is an irreducible dividing
a
1
, then
p
divides all of the
a
i
and
exactly
s
of the
b
i
. Hence
s
±
t
. Reversing the role of the two decom
positions, we get
t
±
s
. Thus the number of direct summands in the two
decompositions is the same.
Fix an irreducible
p
dividing
a
1
. Then
p
divides
a
j
and
b
j
for
1
±
j
±
s
. Let
k
0
be the last index such that
a
k
0
=p
is a unit. Then
pA
j
is
cyclic of period
a
j
=p
for
j > k
0
, while
pA
j
D f
0
g
for
j
±
k
0
, and
pM
D
pA
k
0
C
1
˚ ²²² ˚
pA
s
. Likewise, let
k
00
be the last index such that
b
k
00
=p
is a unit. Then
pB
j
is cyclic of period
b
j
=p
for
j > k
00
, while
pB
j
D f
0
g
for
j
±
k
00
, and
pM
D
pB
k
00
C
1
˚ ²²² ˚
pB
s
.
Applying the induction hypothesis to
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 Fall '08
 EVERAGE
 Algebra, Vector Space

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