On going-down for simple overrings. III

Abstract:

Theorem 1. Let $R$ be an integral domain with quotient field $K$. The following three conditions are equivalent: (a) $R \subset R[u]$ satisfies going-down $(GD)$ for each $u$ in $K$; (b) $R \subset V$ satisfies $GD$ for each valuation overring $V$ of $R$; (c) $R \subset S$ satisfies $GD$ for each domain $S$ containing $R$. If (d) is the condition obtained by restricting the domains $S$ in $({\text {c)}}$ to be overrings of $R$, then $({\text {a)}} \Leftrightarrow {\text {(d)}}$ has been proved in case $R$ is Krull or integrally closed finite-conductor (e.g., pseudo-Bézout) or Noetherian. Theorem 2. Let $R \subset T$ be domains such that either $\operatorname {Spec} (R)$ or $\operatorname {Spec} (T)$, as a poset under inclusion, is a tree. If $R \subset R[u,v]$ satisfies $GD$ for each $u$ and $v$ in $T$, then $R \subset T$ satisfies $GD$.