Journal of Commutative Algebra

Commutative rings whose prime ideals are radically perfect

V. Erdoğdu and S. Harman

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Abstract

The main objective of this paper is to relate the height and the number of generators of ideals in rings that are not necessarily Noetherian. As in [{\bf10, 11}], we call an ideal $I$ of a ring $R$ radically perfect if among the ideals of $R$ whose radical is equal to the radical of $I$ the one with the least number of generators has this number of generators equal to the height of $I$. This is a generalization of the notion of set theoretic complete intersection of ideals in Noetherian rings to rings that need not be Noetherian. In this work, we determine conditions on a ring $R$ so that the prime ideals of $R$ and also those of the polynomial rings $R[X]$ over $R$ are radically perfect. In many cases, it is shown that the condition of prime ideals of $R$ or that of $R[X]$ being radically perfect is equivalent to a form of the class group of $R$ being torsion.

Article information

Source
J. Commut. Algebra, Volume 5, Number 4 (2013), 527-544.

Dates
First available in Project Euclid: 31 January 2014

Permanent link to this document
https://projecteuclid.org/euclid.jca/1391192655

Digital Object Identifier
doi:10.1216/JCA-2013-5-4-527

Mathematical Reviews number (MathSciNet)
MR3161745

Zentralblatt MATH identifier
1282.13016

Subjects
Primary: 13B25: Polynomials over commutative rings [See also 11C08, 11T06, 13F20, 13M10] 13B30: Rings of fractions and localization [See also 16S85] 13C15: Dimension theory, depth, related rings (catenary, etc.) 13C20: Class groups [See also 11R29] 13F05: Dedekind, Prüfer, Krull and Mori rings and their generalizations 13F20: Polynomial rings and ideals; rings of integer-valued polynomials [See also 11C08, 13B25]
Secondary: 13A15: Ideals; multiplicative ideal theory 13A18: Valuations and their generalizations [See also 12J20] 14H50: Plane and space curves

Keywords
Radically perfectness coprime packedness polynomial rings Hilbert domains Krull domains Prüfer domains

Citation

Erdoğdu, V.; Harman, S. Commutative rings whose prime ideals are radically perfect. J. Commut. Algebra 5 (2013), no. 4, 527--544. doi:10.1216/JCA-2013-5-4-527. https://projecteuclid.org/euclid.jca/1391192655


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