If
M
1
,
.
.
.
,
M
s
{M_1},\,...\,,\,{M_s}
are maximal ideals of a ring R that have isomorphic residue fields, then they can be “glued” in the sense that a subring D of R with R is integral over D and
M
1
∩
D
=
.
.
.
=
M
s
∩
D
{M_1}\, \cap \,D\, = \,...\, = \,{M_s}\, \cap \,D
can be constructed. We use this gluing process to prove the following result: Given any finite ordered set
B
\mathcal {B}
, there exists a reduced Noetherian ring B and an embedding
ψ
:
B
→
S
p
e
c
B
\psi :\,\mathcal {B}\, \to \,Spec\,B
such that
ψ
\psi
establishes a bijection between the maximal (respectively minimal) elements of
B
\mathcal {B}
and the maximal (respectively minimal) prime ideals of B and such that given any elements
β
′
\beta ’
,
β
\beta
of
B
\mathcal {B}
, there exists a saturated chain of prime ideals of length r between
ψ
(
β
′
)
\psi (\beta ’)
and
ψ
(
β
)
\psi (\beta )
if and only if there exists a saturated chain of length r between
β
′
\beta ’
and
β
\beta
. We also use the gluing process to construct a Noetherian domain A with quotient field L and a Noetherian domain B between A and L such that:
A
↪
B
A\,\hookrightarrow \,B
possesses the Going Up and the Going Down properties,
A
[
X
]
↪
B
[
X
]
A[X]\,\hookrightarrow \,B[X]
is unibranched and
A
[
X
]
↪
B
[
X
]
A[X]\,\hookrightarrow \,B[X]
possesses neither the Going Up nor the Going Down properties.