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Primary Ideals and Prime Power Ideals

Published online by Cambridge University Press:  20 November 2018

H. S. Butts  and
Robert W. Gilmer Jr.
H. S. Butts
Affiliation:
Louisiana State University, Baton Rouge, Louisiana and Florida State University, Tallahassee, Florida
Robert W. Gilmer Jr.
Affiliation:
Louisiana State University, Baton Rouge, Louisiana and Florida State University, Tallahassee, Florida
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This paper is concerned with the ideal theory of a commutative ring R. We say R has Property (α) if each primary ideal in R is a power of its (prime) radical; R is said to have Property (δ) provided every ideal in R is an intersection of a finite number of prime power ideals. In (2, Theorem 8, p. 33) it is shown that if D is a Noetherian integral domain with identity and if there are no ideals properly between any maximal ideal and its square, then D is a Dedekind domain. It follows from this that if D has Property (α) and is Noetherian (in which case D has Property (δ)), then D is Dedekind.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 18 , 1966 , pp. 1183 - 1195
Copyright
Copyright © Canadian Mathematical Society 1966

References

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