Hostname: page-component-586b7cd67f-2brh9 Total loading time: 0 Render date: 2024-11-22T20:45:46.530Z Has data issue: false hasContentIssue false
  • English
  • Français

Almost-Dedekind rings

Published online by Cambridge University Press:  18 May 2009

E. W. Johnson
E. W. Johnson
Affiliation:
The University of Iowa, Iowa City, Iowa
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.

Throughout we assume all rings are commutative with identity. We denote the lattice of ideals of a ring R by L(R), and we denote by L(R)* the subposet L(R) − R.

A classical result of commutative ring theory is the characterization of a Dedekind domain as an integral domain R in which every element of L(R)* is a product of prime ideals (see Mori [5] for a history). This result has been generalized in a number of ways. In particular, rings which are not necessarily domains but which otherwise satisfy the hypotheses (i.e. general ZPI-rings) have been widely studied (see, for example, Gilmer [3]), as have rings in which only the principal ideals are assumed to satisfy the hypothesis (i.e. π-rings).

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 36 , Issue 1 , January 1994 , pp. 131 - 134
Copyright
Copyright © Glasgow Mathematical Journal Trust 1994

References

REFERENCES

1.Anderson, D. D. and Johnson, E. W., A new characterization of Dedekind domains, Glasgow Math. J. 28 (1986), 237–239.CrossRefGoogle Scholar
2.Butts, H. S. and Gilmer, R. W. Jr, Primary ideals and prime power ideals, Canad. J. Math. 18 (1966), 1183–1195.CrossRefGoogle Scholar
3.Gilmer, R., Multiplicative ideal theory (Marcel Decker, 1972).Google Scholar
4.Levitz, K. B., A characterization of general Z.P.I.-rings, Proc. Amer. Math. Soc. 32 (1972), 376–380.Google Scholar
5.Levitz, K. B., A characterization of general Z.P.I.-rings II, Pacific J. Math. 42 (1972), 147–151.CrossRefGoogle Scholar
6.Mori, S., Allgemeine Z.P.I.-Ringe, J. Sci. Hiroshima Univ. Ser. A. 10 (1940), 117–136.Google Scholar