Hostname: page-component-8448b6f56d-m8qmq Total loading time: 0 Render date: 2024-04-24T22:48:59.398Z Has data issue: false hasContentIssue false
  • English
  • Français

SOME REMARKS ON PRINCIPAL PRIME IDEALS

Part of: General commutative ring theory

Published online by Cambridge University Press:  14 September 2010

D. D. ANDERSON  and
SANGMIN CHUN
D. D. ANDERSON*
Affiliation:
Department of Mathematics, The University of Iowa, Iowa City, IA 52242, USA (email: dan-anderson@uiowa.edu)
SANGMIN CHUN
Affiliation:
Department of Mathematics, Seoul National University, Seoul 151-747, Republic of Korea (email: schun@snu.ac.kr)
*
✉For correspondence; e-mail: dan-anderson@uiowa.edu
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.

In this paper we investigate principal prime ideals in commutative rings. Among other things we characterize the principal prime ideals that are both minimal and maximal and characterize the maximal ideals of a polynomial ring that are principal. Our main result is that if (p) is a principal prime ideal of an atomic ring R, then ht(p)≤1.

Keywords

principal prime ideal

MSC classification

Secondary: 13A15: Ideals; multiplicative ideal theory
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 83 , Issue 1 , February 2011 , pp. 130 - 137
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2010

References

[1]Anderson, D. D., ‘A note on minimal prime ideals’, Proc. Amer. Math. Soc. 122 (1994), 13.CrossRefGoogle Scholar
[2]Anderson, D. D. and Chun, S., ‘Irreducible elements in commutative rings with zero divisors’, Houston J. Math., to appear.Google Scholar
[3]Anderson, D. D., Matijevic, J. and Nichols, W., ‘The Krull intersection theorem II’, Pacific J. Math. 86 (1976), 15–22.CrossRefGoogle Scholar
[4]Gilmer, R., Multiplicative Ideal Theory, Queen’s Papers in Pure and Applied Mathematics, 90 (Queen’s University, Kingston, Ontario, 1992).Google Scholar
[5]Kaplansky, I., Commutative Rings, revised edn (University of Chicago Press, Chicago, IL, 1974).Google Scholar
[6]Loper, A., ‘Two Prüfer domain counterexamples’, J. Algebra 221 (1999), 630–643.CrossRefGoogle Scholar