On $\star$-Power Conductor domains

Let $D$ be an integral domain and $\star$ a star operation defined on $D$. We say that $D$ is a $\star$-power conductor domain ($\star$-PCD) if for each pair $a,b\in D\mathrm{\backslash }(0)$ and for each positive integer $n$ we have $D{a}^n\cap D{b}^n=((Da\cap Db{)}^n{)}^{\ast }\mathrm{.}$ We study $\star$-PCDs and characterize them as root closed domains satisfying $((a,b{)}^n{)}^{-1}=(((a,b{)}^{-1}{)}^n{)}^{\star }$ for all nonzero $a,b$ and all natural numbers $n\ge 1$. From this it follows easily that Pr"{u}fer domains are $d$-PCDs (where $d$ denotes the trivial star operation), and $v$ -domains (e.g., Krull domains) are $v$-PCDs, thereby establishing that a $v$ -domain (e.g., a Prufer or Krull domain) is a $\star$ -PCD. We also consider when a $\star$-PCD is completely integrally closed, and this leads to new characterizations of Krulll domains. In particular, we show that a Noetherian domain is a Krull domain if and only if it is a $w$ -PCD.