Hostname: page-component-7bb8b95d7b-wpx69 Total loading time: 0 Render date: 2024-10-01T17:30:15.797Z Has data issue: false hasContentIssue false
  • English
  • Français

Associated Prime Divisors in the Sense of Krull

Published online by Cambridge University Press:  20 November 2018

Richard A. Kuntz
Richard A. Kuntz*
Affiliation:
Monmouth College, West Long Branch, New Jersey
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.

In a recent paper by Douglas Underwood [8] several definitions of “associated prime divisors” were discussed and shown to be unique. In this note we produce a fifth type, which is due to W. Krull, and is found in his classical paper [2] and further discussed by B. Banaschewski in [1]. Historically this characterization considerably predates the other four definitions.

Throughout this note, R denotes a commutative ring with unity, and all ideals and elements are assumed to be in such a ring. We shall let upper case letters, most frequently the beginning of the alphabet, denote ideals and lower case letters, elements of R. On the whole, our terminology will be that of [9]. We do, however, take exception with [9] in two instances, viz.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 24 , Issue 5 , October 1972 , pp. 808 - 818
Copyright
Copyright © Canadian Mathematical Society 1972

References

1. Banaschewski, B., On the components of ideals in commutative rings, Arch. Math. V. 12 (1961), 22–29.Google Scholar
2. Krull, W., Idealtheorie in ringen ohne Endlichkeitsbedingung, Math. Ann. 101 (1929), 729–744.Google Scholar
3. Krull, W., Uber einen Hauptsatz der allgemeinen Idealtheorie, S. B. Heidelberg Akad. Wiss. Abhandl., 2 (1929), 11–16.Google Scholar
4. Krull, W., Uber Laskersche Ringe, Rend. Circ. Mat. Palermo, Ser 2, 7 (1958), 155–165.Google Scholar
5. McCoy, N. H., Rings and ideals, Carus Math. Monographs No. 8 (Mathematical Association of America, Menascha, Wis., 1948).Google Scholar
6. Nagata, M., Some remarks on prime divisors, Memoirs of the College of Science Univ. of Kyoto, Series A, vol. XXXIII, No. 2 (1960), 297–299.Google Scholar
7. Nagata, M., Local rings (Interscience Publishers, New York, 1962).Google Scholar
8. Underwood, D. H., On some uniqueness questions in primary representations of ideals, J. Math. Kyoto Univ., Vol. 9, No. 1 (1969), 69–94.Google Scholar
9. Zariski, O. and Samuel, P., Commutative algebra Vol. I (D. Van Nostrand Company, Inc. Princeton, 1958).Google Scholar