In this paper an example is given of a pair of commutative noetherian rings
R
⊆
S
R \subseteq S
with
S
S
a finite
R
R
-module and
I
S
∩
R
=
I
IS \cap R = I
for each ideal
I
I
of
R
R
, but having the property that
0
→
R
→
S
0 \to R \to S
is not a pure sequence of
R
R
-modules. Purity of the sequence
0
→
R
→
S
0 \to R \to S
is equivalent to
R
[
X
]
R[X]
being “ideally closed” in
S
[
X
]
,
X
S[X],\;X
an indeterminate. Therefore, the example renders appealing the proposition that for
R
R
noetherian and
S
S
a noetherian torsion-free
R
R
-algebra containing
R
R
, if
α
S
∩
R
=
α
R
\alpha S \cap R = \alpha R
for each non-zero-divisor
α
ϵ
R
\alpha \epsilon R
, then the extension
R
[
X
]
⊆
S
[
X
]
R[X] \subseteq S[X]
has the same properties. Finally, it is also shown that for
R
R
noetherian and
0
→
R
→
S
0 \to R \to S
pure, with
S
S
an
R
R
-algebra, then
R
[
[
X
1
,
…
,
X
n
]
]
R[[{X_1}, \ldots ,{X_n}]]
is pure in
S
[
[
X
1
,
…
,
X
n
]
]
S[[{X_1}, \ldots ,{X_n}]]
for each positive integer
n
n
.