Let (R, T) be a normal pair of commutative rings (i.e., R ${subseteq}$ T is a unita 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 P be one of the following ring-theoretic properties: going-down ring, extensionally going-down (EGD) ring, locally divided ring. Then R has P if and only if T has P. An example shows that the "if" part of the assertion fails if P is taken to be the "divided domain" property.