Let
A
=
k
[
[
X
,
Y
,
Z
]
]
A = k[[X,Y,Z]]
and
k
[
[
T
]
]
k[[T]]
be formal power series rings over a field
k
k
, and let
n
⩾
4
n \geqslant 4
be an integer such that
n
≢
0
mod
3
n\not \equiv 0\;\bmod \;3
. Let
φ
:
A
→
k
[
[
T
]
]
\varphi :A \to k[[T]]
denote the homomorphism of
k
k
-algebras defined by
φ
(
X
)
=
T
7
n
−
3
,
φ
(
Y
)
=
T
(
5
n
−
2
)
n
\varphi (X) = {T^{7n - 3}},\;\varphi (Y) = {T^{(5n - 2)n}}
, and
φ
(
Z
)
=
T
8
n
−
3
\varphi (Z) = {T^{8n - 3}}
. We put
p
=
Ker
φ
{\mathbf {p}} = \operatorname {Ker} \,\varphi
. Then
R
s
(
p
)
=
⊕
i
⩾
0
p
(
i
)
{R_s}({\mathbf {p}}) = { \oplus _{i \geqslant 0}}{{\mathbf {p}}^{(i)}}
is a Noetherian ring if and only if
ch
k
>
0
\operatorname {ch} \,k > 0
. Hence on Cowsik’s question there are counterexamples among the prime ideals defining space monomial curves, too.