Journal of Commutative Algebra

Commutative rings whose prime ideals are radically perfect

V. Erdoğdu and S. Harman

Full-text: Open access


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

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

First available in Project Euclid: 31 January 2014

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

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


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.

Export citation


  • D.D. Anderson, Globalization of some local properties in Krull domains, Proc. Amer. Math. Soc. 85 (1982), 141-145.
  • D.D. Anderson, T. Dumitrescu and M. Zafrullah, Quasi-Schreier domains II, Comm. Alg. 35 (2007), 2096-2104.
  • P.-J.Cahen, Integer valued polynomials on a subset, Proc. Amer. Math. Soc. 117 (1993), 919-929.
  • D. Eisenbud and E.G. Evans, Every algebraic set in $n$-space is the intersection of n hypersurfaces, Inv. Math. 19 (1973), 107-112.
  • V. Erdoğdu, Coprimely packed rings, J. Number Theor. 28 (1988), 1-5.
  • –––, The prime avoidance of ideals in Noetherian Hilbert rings, Comm. Alg. 22 (1994), 4989-4990.
  • –––, The prime avoidance of maximal ideals in commutative rings, Comm. Alg. 23 (1995), 863-868.
  • –––, Three notes on coprime packedness, J. Pure Appl. Alg. 148 (2000), 165-170.
  • –––, Coprime packedness and set theoretic complete intersections of ideals in polynomial rings, Proc. Amer. Math. Soc. 132 (2004), 3467-3471.
  • –––, Radically perfect prime ideals in polynomial rings, Arch. Math. 93 (2009), 213-217.
  • –––, Efficient generation of prime ideals in polynomial rings up to radical, Comm. Alg. 38 (2010), 1802-1807.
  • V. Erdoğdu and S. McAdam, Coprimely packed Noetherian polynomial rings, Comm. Alg. 22 (1994), 6459-6470.
  • M. Fontana and S. Kabbaj, Essential domains and two conjectures in dimension theory, Proc. Amer. Math. Soc. 132 (2004), 2529-2535.
  • R. Gilmer, Multiplicative ideal theory, Queen's Papers Pure Appl. Math. 90, Queen's University, Kingston Ontario, 1992.
  • O. Goldman, Hilbert rings and the Hilbert Nullstellensatz, Math. Z. 54 (1951), 136-140.
  • I. Kaplansky, Commutative rings, University of Chicago Press, Boston, 1974.
  • E. Kunz, Introduction to commutative algebra and algebraic geometry, Birkhauser, Boston, 1985.
  • G. Lyubeznik, The number of defining equations of affine algebraic sets, Amer. J. Math. 114 (1992), 413-463.
  • H. Matsumura, Commutative ring theory, Cambridge University Press, Cambridge, 1986.
  • R. Pendleton, A characterization of $Q$-domains, Bull. Amer. Math. Soc. 72 (1966), 499-500. \noindentstyle