Abstract
Let
k
be a perfect field of characteristic
p>0
,
k(t)_{per}
the perfect closure of
k(t)
and
A
a
k
-algebra. We characterize whether the ring
A\otimes_k k(t)_{per}=\bigcup_{m\geq 0}(A\otimes_k k(t^{\frac{1}{p^m}}))
is noetherian or not. As a consequence, we prove that the ring
A\otimes_k k(t)_{per}
is noetherian when
A
is the ring of formal power series in
n
indeterminates over
k
.