Finite generation of powers of ideals

Authors:
Robert Gilmer, William Heinzer and Moshe Roitman

Journal:
Proc. Amer. Math. Soc. **127** (1999), 3141-3151

MSC (1991):
Primary 13A15, 13E99, 13G05

DOI:
https://doi.org/10.1090/S0002-9939-99-05199-0

Published electronically:
May 4, 1999

MathSciNet review:
1646305

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Suppose is a maximal ideal of a commutative integral domain and that some power of is finitely generated. We show that is finitely generated in each of the following cases: (i) is of height one, (ii) is integrally closed and , (iii) is a monoid domain over a field , where is a cancellative torsion-free monoid such that , and is the maximal ideal . We extend the above results to ideals of a reduced ring such that is Noetherian. We prove that a reduced ring is Noetherian if each prime ideal of has a power that is finitely generated. For each with , we establish existence of a -dimensional integral domain having a nonfinitely generated maximal ideal of height such that is -generated.

**1.**Paul Eakin and Avinash Sathaye,*Prestable ideals*, J. Algebra**41**(1976), no. 2, 439–454. MR**0419428**, https://doi.org/10.1016/0021-8693(76)90192-7**2.**Stefania Gabelli,*Completely integrally closed domains and 𝑡-ideals*, Boll. Un. Mat. Ital. B (7)**3**(1989), no. 2, 327–342 (English, with Italian summary). MR**997999****3.**Robert Gilmer,*On factorization into prime ideals*, Comment. Math. Helv.**47**(1972), 70–74. MR**0306183**, https://doi.org/10.1007/BF02566789**4.**Robert W. Gilmer,*Multiplicative ideal theory*, Queen’s Papers in Pure and Applied Mathematics, No. 12, Queen’s University, Kingston, Ont., 1968. MR**0229624****5.**Robert Gilmer,*Commutative semigroup rings*, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1984. MR**741678****6.**William Heinzer, David Lantz, and Kishor Shah,*The Ratliff-Rush ideals in a Noetherian ring*, Comm. Algebra**20**(1992), no. 2, 591–622. MR**1146317**, https://doi.org/10.1080/00927879208824359**7.**Irving Kaplansky,*Commutative rings*, Revised edition, The University of Chicago Press, Chicago, Ill.-London, 1974. MR**0345945****8.**Masayoshi Nagata,*Local rings*, Interscience Tracts in Pure and Applied Mathematics, No. 13, Interscience Publishers a division of John Wiley & Sons New York-London, 1962. MR**0155856****9.**Jack Ohm and R. L. Pendleton,*Rings with noetherian spectrum*, Duke Math. J.**35**(1968), 631–639. MR**0229627****10.**L. J. Ratliff Jr. and David E. Rush,*Two notes on reductions of ideals*, Indiana Univ. Math. J.**27**(1978), no. 6, 929–934. MR**0506202**, https://doi.org/10.1512/iumj.1978.27.27062

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (1991):
13A15,
13E99,
13G05

Retrieve articles in all journals with MSC (1991): 13A15, 13E99, 13G05

Additional Information

**Robert Gilmer**

Affiliation:
Department of Mathematics, Florida State University Tallahassee, Florida 32306-4510

Email:
gilmer@math.fsu.edu

**William Heinzer**

Affiliation:
Department of Mathematics, Purdue University, West Lafayette, Indiana 47907-1395

Email:
heinzer@math.purdue.edu

**Moshe Roitman**

Affiliation:
Department of Mathematics, University of Haifa, Mount Carmel, Haifa 31905, Israel

Email:
mroitman@mathcs2.haifa.ac.il

DOI:
https://doi.org/10.1090/S0002-9939-99-05199-0

Keywords:
Cohen's theorem,
finite generation,
maximal ideal,
monoid ring,
Noetherian,
power of an ideal,
Ratliff-Rush closure

Received by editor(s):
January 26, 1998

Published electronically:
May 4, 1999

Additional Notes:
The first two authors acknowledge with thanks the hospitality of the mathematics department of the University of North Carolina at Chapel Hill. Partial support of the work of the second author by the National Science Foundation is also gratefully acknowledged.

Communicated by:
Wolmer V. Vasconcelos

Article copyright:
© Copyright 1999
American Mathematical Society