Let
D
D
and
J
J
be integral domains such that
D
⊂
J
D \subset J
and
J
[
[
X
]
]
J[[X]]
is not algebraic over
D
[
[
X
]
]
D[[X]]
. Is it necessarily the case that there exists an integral domain
R
R
such that
D
[
[
X
]
]
⊂
R
⊆
J
[
[
X
]
]
D[[X]] \subset R \subseteq J[[X]]
and
R
≅
D
[
[
X
]
]
[
[
{
Y
i
}
i
=
1
∞
]
]
R \cong D[[X]][[\{ {Y_i}\} _{i = 1}^\infty ]]
? While the general question remains open, the question is answered affirmatively in a number of cases. For example, if
D
D
satisfies any one of the conditions (1)
D
D
is Noetherian, (2)
D
D
is integrally closed, (3) the quotient field
K
K
of
D
D
is countably generated as a ring over
D
D
, or (4)
D
D
has Krull dimension one, then an affirmative answer is given. Further, in the Noetherian case it is shown that
J
[
[
X
]
]
J[[X]]
is algebraic over
D
[
[
X
]
]
D[[X]]
if and only if it is integral over
D
[
[
X
]
]
D[[X]]
and necessary and sufficient conditions are given on
D
D
and
J
J
in order that this occur. Finally if, for every positive integer
n
n
,
D
[
[
X
1
,
…
,
X
n
]
]
⊂
R
⊆
J
[
[
X
1
,
…
,
X
n
]
]
D[[{X_1}, \ldots ,{X_n}]] \subset R \subseteq J[[{X_1}, \ldots ,{X_n}]]
implies that
R
≆
D
[
[
X
1
,
…
,
X
n
]
]
[
[
{
Y
i
}
i
=
1
∞
]
]
R \ncong D[[{X_1}, \ldots ,{X_n}]][[\{ {Y_i}\} _{i = 1}^\infty ]]
, then it is shown that
J
[
[
X
1
,
…
,
X
n
]
]
J[[{X_1}, \ldots ,{X_n}]]
is algebraic over
D
[
[
X
1
,
…
,
X
n
]
]
D[[{X_1}, \ldots ,{X_n}]]
for every
n
n
.