  We study the effects on $D$ of assuming that the power series ring $D[[X]]$
is a $v$-domain or a PVMD. We show that a PVMD $D$ is completely integrally
closed if and only if $\bigcap_{n=1}^{\infty }(I^{n})_{v}=(0)$ for every proper
$t$-invertible $t$-ideal $I$ of $D$. Using this, we show that if $D$ is an AGCD
domain, then $D[[X]]$ is integrally closed if and only if $D$ is a completely
integrally closed PVMD with torsion $t$-class group. We also determine several
classes of PVMDs for which being Archimedean is equivalent to being completely
integrally closed and give some new characterizations of integral domains
related to Krull domains.
