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A GENERALIZATION OF PRÜFER'S ASCENT RESULT TO NORMAL PAIRS OF COMPLEMENTED RINGS

DAVID E. DOBBS

Department of Mathematics, University of Tennessee, Knoxville, TN 37996–1320, USA

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and

JAY SHAPIRO

Department of Mathematics, George Mason University, Fairfax, VA 22030–4444, USA

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https://doi.org/10.1142/S021949881100521XCited by:4 (Source: Crossref)

Abstract

Let R ⊆ T be a (unital) extension of (commutative) rings, such that the total quotient ring of R is a von Neumann regular ring and T is torsion-free as an R-module. Let T ⊆ B be a ring extension such that B is a reduced ring that is torsion-free as a T-module. Let R* (respectively, A) be the integral closure of R in T (respectively, in B). Then (R*, T) is a normal pair (i.e. S is integrally closed in T for each ring S such that R* ⊆ S ⊆ T) if and only if (A, AT) is a normal pair. This generalizes results of Prüfer and Heinzer on Prüfer domains to normal pairs of complemented rings.

Keywords:

AMSC: 13B22, 13B21, 13A15, 13C13, 13G05, 13F05

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