Math 5126
Monday, February 4
Third Homework Solutions
1. First we write
(
x
+
1
)(
x
3
+
1
)
as a product of irreducibles, namely
(
x
+
1
)
2
(
x
2

x
+
1
)
; it is easy to see that
x
2

x
+
1 is irreducible over
Q
(note that the roots are not
real). If
R
is a PID and
x
=
p
e
1
1
...
p
e
n
n
where the
p
i
are nonassociate primes, then a
ﬁnitely generated
R
/
(
x
)
module is isomorphic to a direct sum of modules of the form
R
/
(
p
f
1
1
...
p
f
n
n
)
, where
f
i
≤
e
i
, and such a module is projective if and only if
f
i
=
e
i
for
all
i
. It follows that the ﬁnitely generated projective
Q
[
x
]
/
(
x
+
1
)(
x
3
+
1
)
modules are
direct sums of modules of the form
Q
[
x
]
/
(
x
+
1
)
2
and
Q
[
x
]
/
(
x
2

x
+
1
)
.
2. By the structure theorem for ﬁnitely generated modules over a PID,
M
∼
=
R
n
⊕
T
,
where
T
is a torsion module. Since
N
is not a torsion module, we see that
M
6
=
T
and
hence
n
6
=
0. We deduce that
M
∼
=
R
⊕
X
for some
R
module
X
(where
X
∼
=
R
n

1
⊕
T
)
and the ﬁrst statement is proven.
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 Fall '07
 PALinnell
 Math, Algebra, Determinant, Abelian group, Module theory, Structure theorem for finitely generated modules over a principal ideal domain, smith normal form

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