Hostname: page-component-586b7cd67f-rcrh6 Total loading time: 0 Render date: 2024-11-27T22:30:05.749Z Has data issue: false hasContentIssue false
  • English
  • Français

Distinguished Domains

Published online by Cambridge University Press:  20 November 2018

Raymond C. Heitmann  and
Stephen McAdam
Raymond C. Heitmann
Affiliation:
The University of Texas, Austin, Texas
Stephen McAdam
Affiliation:
The University of Texas, Austin, Texas
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

This paper introduces a class of domains which we hope to show merits some attention.

Definition. The domain R is said to be a distinguished domain if for any 0 ≠ z ∈ K, the quotient field of R, (1 : z) does not consist entirely of zero divisors modulo (1 : z–l). (Note: Here we use the fact that a zero module has no zero divisors. Thus if z–l ∈ R, so that (1 : z–l) = R, then the condition holds trivially.)

Section 1 of this paper gives numerous examples of distinguished domains, foremost among them being Krull domains and Prufer domains. In fact Prüfer domains are shown to be exactly those distinguished domains whose prime lattice forms a tree. Other distinguished domains can be constructed by the D + M construction. It is shown that distinguished domains are integrally closed but the converse fails.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 34 , Issue 1 , 01 February 1982 , pp. 181 - 193
Copyright
Copyright © Canadian Mathematical Society 1982

References

1. Eakin, P. and Silver, J., Rings which are almost polynomial rings, Trans. AMS 17J+ (1972), 425–449.Google Scholar
2. Nagata, M., Local rings (Interscience, New York, 1962).Google Scholar