# Normal Pairs of Going-down Rings

Dobbs, David Earl;Shapiro, Jay Allen

• Accepted : 2010.12.09
• Published : 2011.03.31
• 26 17

#### Abstract

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.

#### 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

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2. On Finite Maximal Chains of Weak Baer Going-Down Rings vol.40, pp.5, 2012, https://doi.org/10.1080/00927872.2011.558881
3. A GENERALIZATION OF PRÜFER'S ASCENT RESULT TO NORMAL PAIRS OF COMPLEMENTED RINGS vol.10, pp.06, 2011, https://doi.org/10.1142/S021949881100521X