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PACIFIC JOURNAL OF MATHEMATICS
Vol. 15, No. 1. 1965
DEDEKIND
DOMAINS
AND RINGS OF
QUOTIENTS
LUTHER CLABORN
We study
the relation of the ideal class group of a
Dedekind domain
A
to that of
A
s
,
where
S is a multiplicatively
closed subset
of
A.
We construct examples of (a) a Dedekind
domain with
no principal prime ideal and (b) a Dedekind
domain which
is not the integral closure of a principal ideal
domain.
We also obtain some qualitative information on the
number
of nonprincipal prime ideals in an arbitrary Dedekind
If
A is a Dadekind domain,
S
the set of all monic poly
nomials
and
T
the set of all primitive polynomials of
A[X],
then A[X]<? and
A[X]
T
are both Dadekind domains. We obtain
the class groups
of these new Dsdekind domains in terms of
that
A.
1* LEMMA
11.
If A is a Dedekind domain and S is a multi
plicatively closed
set of A suoh that A
s
is not a field, then A
s
is
also
a Dedekind domain.
Proof.
That
A
s
is integrally closed and Noetherian if
A
is, follows
from
the general theory of quotient ring formations. The primes of
A
s
are of the type
PA
S)
where
P
is a prime ideal of
A
such that
PΓ)S
= ψ.
Since height
PA
S
= height
P
if
PΠS = φ, P Φ
(0) and
PΠS
= φ
imply that height
PA
S
=
1.
PROPOSITION
12. If
A
is a Dedekind domain and
S
is a multi
plicatively closed
set of
A,
the assignment
C
—
»
CA
S
is a mapping of
the
set of fractionary ideals of
A
onto
the set of fractionary ideals
A
s
which
is a homomorphism for multiplication.
Proof.
C
is a fractionary ideal of
A
if and only if there is a
d
G
A
dC
S
A.
If this is so, certainly
dCA
s
S
A
Sj
so
CA
S
is
a fractionary ideal of
A
s
.
Clearly
(J5
C)A
S
~ BA
S
CA
S
,
so the
assignment
is a homomorphism. Let
D
be any fractionary ideal of
A
s
.
Since
A
s
is a Dedekind domain,
D
is in the free group generated
by
all prime ideals of
A
s
,
i.e.
D = Q

1
Ql
k
.
For each
i —
1, ,
k
there
is a prime
P
i
A
Q
i
= P
{
A
Sa
Set
E
=
Pi
1
P
n
k
K
Then using
the fact that we have a multiplicative homomorphism of
fractionary ideals,
we get
Received December 13, 1963.
59
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LUTHER CLABORN
EA
a
= (P
1
A
S
)^
(P
k
A
s
y*
= Q i .
..
QlK
COROLLARY
13.
Let A be a Dedekind domain and S be a multi
plicatively
closed
set of A. Let C (for C a
fractionary ideal
of A
or
A
s
)
denote
the
class
of the ideal
class
group
to
which
C
belongs.
Then
the
assignment
C
—
*
CA
S
is a
homomorphism
φ of the ideal
class
group
of A
onto that
of A
s
.
Proof
It is only necessary to note that if
C
=
dA,
then
CA
S
=
dA
s
.
THEOREM
14.
The kernel of φ is
generated
by all P
ω
,
where
Pa
ranges
over
all
primes
such that P#OS
Φ Φ.
If
P
a
ΠS
Φ φ,
then
P^As —
A
s
.
Suppose
C
is a fractionary ideal
such that
C
=
P
ay
i.e.
C
=
dP
ω
for some
d
in the quotient field of
A.
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 Fall '11
 Clark
 Geometry

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