QUALIFYING EXAMINATION
Math 554
January 2005  Prof. Ulrich
1.
(12 points)
Without proof give an answer to these questions:
(a) For
R
a commutative ring and
M
an
R
module, is the
R
module Hom
R
(
M, R
)
torsion free?
(b) How many isomorphism classes are there of
Z
20
modules having exactly
625 elements?
(c) How many isometry classes are there of alternating bilinear forms on a
3dimensional vector space?
2.
(15 points)
Let
R
be a commutative ring and
P
an
R
module. Prove that the following are
equivalent:
(a) For every
R
linear map
f
:
P
→
N
and every
R
epimorphism
π
:
M
→
N
there exists an
R
linear map
g
:
P
→
M
with
πg
=
f
.
(b) There exists an
R
module
Q
such that the direct sum
P
⊕
Q
is a free
R
module.
3.
(15 points)
Let
R
be an integral domain and
F
a free
R
module with ordered basis
{
x
1
, . . . , x
n
}
.
Let
M
=
Ru
1
+
. . .
+
Ru
n
⊂
N
=
Rv
1
+
. . .
+
Rv
n
be submodules of
F
with
u
i
=
∑
j
a
ij
x
j
,
v
i
=
∑
j
b
ij
x
j
(
a
ij
, b
ij
∈
R
), and consider the
n
by
n
matrices
A
= (
a
ij
),
B
= (
b
ij
).
(a) Prove that
M
and
N
are free
R
modules if the determinant det
A
6
= 0.
(b) Assume det
A
6
= 0. Prove that
M
=
N
if and only if det
A
and det
B
are
associates.
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4.
(13 points)
Determine whether the matrices
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 Spring '09
 Math, Vector Space, Ring, Commutative ring

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