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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
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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
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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
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6.Mori, S., Allgemeine Z.P.I.-Ringe, J. Sci. Hiroshima Univ. Ser. A. 10 (1940), 117–136.Google Scholar