Let
A
A
be a Noetherian Cohen-Macaulay domain,
b
b
,
c
1
c_1
,
…
\dots
,
c
g
c_g
an
A
A
-sequence,
J
J
=
(
b
,
c
1
,
…
,
c
g
)
A
(b,c_1,\dots ,c_g)A
, and
B
B
=
A
[
J
/
b
]
A[J/b]
. Then
B
B
is Cohen-Macaulay, there is a natural one-to-one correspondence between the sets
Ass
B
(
B
/
b
B
)
\operatorname {Ass}_B(B/bB)
and
Ass
A
(
A
/
J
)
\operatorname {Ass}_A(A/J)
, and each
q
q
∈
\in
Ass
A
(
A
/
J
)
\operatorname {Ass}_A(A/J)
has height
g
+
1
g+1
. If
B
B
does not have unique factorization, then some height-one prime ideals
P
P
of
B
B
are not principal. These primes are identified in terms of
J
J
and
P
∩
A
P \cap A
, and we consider the question of how far from principal they can be. If
A
A
is integrally closed, necessary and sufficient conditions are given for
B
B
to be integrally closed, and sufficient conditions are given for
B
B
to be a UFD or a Krull domain whose class group is torsion, finite, or finite cyclic. It is shown that if
P
P
is a height-one prime ideal of
B
B
, then
P
∩
A
P \cap A
also has height one if and only if
b
b
∉
\notin
P
P
and thus
P
∩
A
P \cap A
has height one for all but finitely many of the height-one primes
P
P
of
B
B
. If
A
A
has unique factorization, a description is given of whether or not such a prime
P
P
is a principal prime ideal, or has a principal primary ideal, in terms of properties of
P
∩
A
P \cap A
. A similar description is also given for the height-one prime ideals
P
P
of
B
B
with
P
∩
A
P \cap A
of height greater than one, if the prime factors of
b
b
satisfy a mild condition. If
A
A
is a UFD and
b
b
is a power of a prime element, then
B
B
is a Krull domain with torsion class group if and only if
J
J
is primary and integrally closed, and if this holds, then
B
B
has finite cyclic class group. Also, if
J
J
is not primary, then for each height-one prime ideal
p
p
contained in at least one, but not all, prime divisors of
J
J
, it holds that the height-one prime
p
A
[
1
/
b
]
∩
B
pA[1/b] \cap B
has no principal primary ideals. This applies in particular to the Rees ring
R
{\mathbf R}
=
=
A
[
1
/
t
,
t
J
]
A[1/t, tJ]
. As an application of these results, it is shown how to construct for any finitely generated abelian group
G
G
, a monoidal transform
B
B
=
A
[
J
/
b
]
A[J/b]
such that
A
A
is a UFD,
B
B
is Cohen-Macaulay and integrally closed, and
G
G
≅
\cong
Cl
(
B
)
\operatorname {Cl}(B)
, the divisor class group of
B
B
.