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Dedekind Domains with Torsion Class Group
David Krumm
Let
R
be a Dedekind domain with fraction ﬁeld
K
. An
overring
of
R
is any ring
A
such that
R
⊆
A
⊆
K
.
I will prove the following result:
The class group of
R
is a torsion group if and only if every overring of
R
is a localization of
R
.
Suppose ﬁrst that the class group of
R
is torsion, and let
A
be any overring of
R
. We know that
A
=
\
p
∈
T
R
p
(
?
)
for some set
T
of prime ideals of
R
. Explicitly,
T
=
{
p
:
p
A
6
=
A
}
. I will show that
A
=
S

1
R
where
S
is
the multiplicative set
R
\ ∪
p
∈
T
p
. It is clear from the deﬁnition of localization that
S

1
R
⊆
A
. To see the
reverse inclusion, ﬁx an element
a
∈
A
and consider the ideal
D
=
{
r
∈
R
:
ra
∈
R
}
of denominators of
a
. Since the class group of
R
is torsion, there is a positive integer
n
such that
D
n
is a principal ideal, say
D
n
= (
s
). Thus for any prime ideal
p
of
R
we have that
p
divides
D
if and only if
s
∈
p
. Since
D
intersects
R
\
p
for all
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 Fall '11
 Staff
 Number Theory

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