Semistar normal pairs and related characterizations of P*MDs

چکیده مقاله

Given an overring extension of domains $D \subseteq
T$, we introduce the semistar-theoretic analogues of $D$ being
integrally closed in $T$, $(D,T)$ being a normal pair, $D
\subseteq T$ being a residually algebraic extension and $(D,T)$
being a residually algebraic pair. Our main result generalizes
Davis' characterization of normal pairs $(D,T)$ as (overring)
extensions $D \subseteq T$ for which each intermediate ring is
$D$-flat, while also generalizing some celebrated characterizations of Pr\"ufer domains and several results of
Ayache-Jabbalah on residually algebraic pairs. This work also involves extending the connections between the semistar-theoretic notions of primitivity and $\INC$-pair that were given by Chang and Fontana. Applications
include several characterizations of P$\star$MDs.