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8.5. FINITELY GENERATED MODULES OVER A PID, PART II.
381
(b)
The decomposition in part (a) is unique, in the following sense:
Suppose
M
Š
R=.b
1
/
˚
R=.b
2
/
˚ ±±± ˚
R=.b
t
/
˚
R
`
;
where the
b
i
are nonzero, nonunit elements of
R
, and
b
i
divides
b
j
for
i
²
j
. Then
s
D
t
,
`
D
k
and
.a
i
/
D
.b
i
/
for all
i
.
Before addressing the uniqueness statement in the theorem, we intro
duce the idea of
torsion
.
Suppose that
R
is an integral domain (not necessarily a PID) and
M
is an
R
–module.
For
x
2
M
, recall that the
annihilator
of
x
in
R
is
ann
.x/
D f
r
2
R
W
rx
D
0
g
, and that ann
.x/
is an ideal in
R
. Since
1x
D
x
, ann
.x/
³
¤
R
. Recall from Example
8.2.6
that
Rx
Š
R=
ann
.x/
as
R
–modules. An element
x
2
M
is called a
torsion element
if ann
.x/
¤
f
0
g
, that is, there exists a nonzero
r
2
R
such that
rx
D
0
.
If
x;y
2
M
are two torsion elements then
sx
C
ty
is also a torsion
element for any
s;t
2
R
. In fact, if
r
1
is a nonzero element of
R
such
that
r
1
x
D
0
and
r
2
is a nonzero element of
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 Fall '08
 EVERAGE
 Algebra

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