Let $(R,T)$ be a normal pair of commutative rings (i.e., $R subseteq T$ is a unital extension of commutative rings, not necessarily integral domains, such that $S$ is integrally closed in $T$ for each ring $S$ such that $R subseteq S subseteq T$) such that the total quotient ring of $R$ is a von Neumann regular ring. Let $mathcal{P}$ be one of the following ring-theoretic properties: going-down ring, extensionally going-down (EGD) ring, locally divided ring. Then $R$ has $mathcal{P}$ if and only if $T$ has $mathcal{P}$. An example shows that the ``if" part of the assertion fails if $mathcal{P}$ is taken to be the ``divided domain" property.

Keywords: Normal pair, prime ideal, total quotient ring, valuation domain, divided domain, pullback, going-down ring, EGD ring, locally divided ring, weak Baer ring, reduced ring