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On Chain Conditions in Integral Domains

Published online by Cambridge University Press:  20 November 2018

Valentina Barucci  and
David E. Dobbs
Valentina Barucci
Affiliation:
Istituto Matematico, Università Di Roma I00185 Roma, Italy
David E. Dobbs
Affiliation:
Department of Mathematics, University of TennesseeKnoxville, Tennessee 37996, U.S.A.
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Abstract

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The following two theorems are proved. If R is an Archimedean conducive integral domain, then R is quasilocal and dim(R) ≤1. If each overring of an integral domain R has ascending chain condition on divisorial ideals, then the integral closure of R is a Dedekind domain. Both theorems sharpen results already known in the Noetherian case. The second theorem leads to a strengthened converse of the Krull-Akizuki Theorem. We also investigate the effect of restricting the hypothesis in the second theorem to the proper overrings of R.

Keywords

13E0513F0513B2013G05
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 27 , Issue 3 , 01 September 1984 , pp. 351 - 359
Copyright
Copyright © Canadian Mathematical Society 1984

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