Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-26T19:46:07.543Z Has data issue: false hasContentIssue false
  • English
  • Français

On Inc-Extensions and Polynomials with Unit Content

Published online by Cambridge University Press:  20 November 2018

David E. Dobbs
David E. Dobbs*
Affiliation:
Department of Mathematics, Ayres Hall, University of Tennessee, Knoxville, Tennessee 37916
Rights & Permissions [Opens in a new window]

Abstract

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.

It is proved that if u is an element of a faithful algebra over a commutative ring R, then u satisfies a polynomial over R which has unit content if and only if the extension R ⊂ R[u] has the imcomparability property. Applications include new proofs of results of Gilmer-Hoffmann and Papick, as well as a characterization of the P-extensions introduced by Gilmer and Hoffmann.

Keywords

13A1513F0513B2013B25Prime idealp-extensionincomparability propertyintegral closurePriifer domainlying-over propertygoing-up propertycoherent ring
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 23 , Issue 1 , 01 March 1980 , pp. 37 - 42
Copyright
Copyright © Canadian Mathematical Society 1980

References

1. Dawson, J. and Dobbs, D. E., On going down in polynomial rings, Canad. J. Math. 26 (1974), 177-184.Google Scholar
2. Dobbs, D. E., On going down for simple overrings, II, Comm. in Algebra 1 (1974), 439-458.Google Scholar
3. Evans, E. G. Jr., A generalization of ZariskVs main theorem, Proc. Amer. Math. Soc. 26 (1970), 45-48.Google Scholar
4. Gilmer, R. and Hoffmann, J. F., A characterization of Prufer domains in terms of polynomials, Pac. J. Math. 60 (1975), 81-85.Google Scholar
5. Kaplansky, I., Going up in polynomial rings, unpublished manuscript, 1972.Google Scholar
6. Kaplansky, I., Commutative rings, rev. ?d., University of Chicago Press, Chicago and London, 1974.Google Scholar
7. McAdam, S., Going down in polynomial rings, Canad. J. Math. 23 (1971), 704-711.Google Scholar
8. Papick, I. J., Topologically defined classes of going-down domains, Trans. Amer. Math. Soc. 219 (1976), 1-37.Google Scholar
9. Papick, I. J., Coherent overrings, Canad. Math. Bull., to appear.Google Scholar
10. Seidenberg, A., A note on the dimension theory of rings, Pac. J. Math. 3 (1953), 505-512.Google Scholar