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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 I 00185 Roma, Italy
David E. Dobbs
Affiliation:
Department of Mathematics, University of Tennessee Knoxville, 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

Information

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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