Normal Pairs of Going-down Rings

Dobbs, David Earl;Shapiro, Jay Allen

  • Received : 2010.06.04
  • Accepted : 2010.12.09
  • Published : 2011.03.31


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.


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


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