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QUALIFYING EXAM
Math 554
August 2003
1.
(12 points) Without proof give an example of:
(a) a domain
R
and a ﬁnitely generated torsionfree
R
module that is not free;
(b) a nonzero commutative ring
R
that is not a ﬁeld so that every
R
module
is torsionfree;
(c) a normal matrix with entries in
R
that is not similar to a diagonal matrix
with entries in
R
.
2.
(13 points) For
R
a commutative ring and
M
an
R
module let
M
?
=Hom
R
(
M,R
)
denote the
R
dual of
M
.
(a) Let
M
and
N
be
R
modules. Show that (
M
⊕
N
)
?
∼
=
M
?
⊕
N
?
.
(b) Let
R
be a principal ideal domain and
M
a ﬁnitely generated
R
module.
Show that
M
?
is a free
R
module with rank
M
?
=rank
M
.
3.
(12 points) Let
Q
[
X
] be the polynomial ring in one variable over
Q
,
n
a positive
integer, and
R
=
Q
[
X
]
/
(
X
n
). Classify all ﬁnitely generated
R
modules up to
R
isomorphisms.
4.
(15 points) Let
R
be a commutative ring,
F
afree
R
module of ﬁnite rank, and
ϕ
∈
End
R
(
F
)an
R
endomorphism of
F
. Show that the following are equivalent:
(a)
ϕ
is bijective;
(b)
ϕ
is surjective;
(c) det(
ϕ
) is a unit of
R
.
5.
(20 points) Let
K
be a ﬁeld,
V
a
K
vector space of dimension
n
,End
K
(
V
)
the
K
vector space of
K
endomorphisms of
V
,and
ϕ
∈
End
K
(
V
)aﬁxed
K

endomorphism. Show that:
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 Spring '09
 Math

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