If
$A$
is a subring of a commutative ring
$B$
and if
$n$
is a positive integer, a number of sufficient conditions are given for “
$A[[X]]$
is
$n$
-root closed in
$B[[X]]$
” to be equivalent to “
$A$
is
$n$
-root closed in
$B$
.” In addition, it is shown that if
$S$
is a multiplicative submonoid of the positive integers
$\mathbb{P}$
which is generated by primes, then there exists a one-dimensional quasilocal integral domain
$A$
(resp., a von Neumann regular ring
$A$
) such that
$S=\{n\in \mathbb{P}|A\,\,\text{is}\,n-\text{root}\,\text{closed}\}$
(resp.,
$S=\{n\in \mathbb{P}\,|\,\,A[[X]]\,\,\text{is}\,n-\text{root}\,\text{closed}\}$
).