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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

1. Butts, H. S. and Phillips, R. C., Almost multiplication rings, Can. J. Math., 17 (1965), 267–277.Google Scholar
2. Cohen, I. S., Commutative rings with restricted minimum condition, Duke Math. J., 17 (1950), 27–42.Google Scholar
3. Gilmer, R., A class of domains in which primary ideals are valuation ideals, Math. Ann., 161 (1965), 247–254.Google Scholar
4. Gilmer, R., Eleven nonequivalent conditions on a commutative ring, Nagoya Math. J., 26 (1966), 183–194.Google Scholar
5. Gilmer, R., Extension of results concerning rings in which semi-primary ideals are primary, Duke Math. J., 31 (1964), 73–78.Google Scholar
6. Gilmer, R., On a classical theorem of Noether in ideal theory, Pac. J. Math., 18 (1963), 579–583.Google Scholar
7. Gilmer, R., The cancellation law for ideals of a commutative ring, Can J. Math., 17 (1965), 281–287.Google Scholar
8. Gilmer, R. and Mott, J., Multiplication rings as rings in which ideals with prime radical are primary, Trans. Amer. Math. Soc., 114 (1965), 40–52.Google Scholar
9. Gilmer, R. and Ohm, J., Primary ideals and valuation ideals, Trans. Amer. Math. Soc., 117 (1965), 237–250.Google Scholar
10. Mori, S., Allgemeine Z.P.I.-Ringe, J. Sci. Hiroshima Univ., Ser. A, 10 (1940), 117–136.Google Scholar
11. van der Waerden, B. L., Modern algebra, vol. I (New York, 1949).Google Scholar
12. Zariski, O. and Samuel, P., Commutative algebra, vol. I (Princeton, 1958).Google Scholar
13. Zariski, O. and Samuel, P., Commutative algebra, vol. II (Princeton, 1960).Google Scholar