Finite generation of powers of ideals
Authors:
Robert Gilmer, William Heinzer and Moshe Roitman
Journal:
Proc. Amer. Math. Soc. 127 (1999), 31413151
MSC (1991):
Primary 13A15, 13E99, 13G05
Published electronically:
May 4, 1999
MathSciNet review:
1646305
Fulltext 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 torsionfree 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
(54 #7449)
 2.
Stefania
Gabelli, Completely integrally closed domains and
𝑡ideals, Boll. Un. Mat. Ital. B (7) 3
(1989), no. 2, 327–342 (English, with Italian summary). MR 997999
(90h:13019)
 3.
Robert
Gilmer, On factorization into prime ideals, Comment. Math.
Helv. 47 (1972), 70–74. MR 0306183
(46 #5310)
 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
(37 #5198)
 5.
Robert
Gilmer, Commutative semigroup rings, Chicago Lectures in
Mathematics, University of Chicago Press, Chicago, IL, 1984. MR 741678
(85e:20058)
 6.
William
Heinzer, David
Lantz, and Kishor
Shah, The RatliffRush ideals in a Noetherian ring, Comm.
Algebra 20 (1992), no. 2, 591–622. MR 1146317
(93c:13002), http://dx.doi.org/10.1080/00927879208824359
 7.
Irving
Kaplansky, Commutative rings, Revised edition, The University
of Chicago Press, Chicago, Ill.London, 1974. MR 0345945
(49 #10674)
 8.
Masayoshi
Nagata, Local rings, Interscience Tracts in Pure and Applied
Mathematics, No. 13, Interscience Publishers a division of John Wiley &
Sons New YorkLondon, 1962. MR 0155856
(27 #5790)
 9.
Jack
Ohm and R.
L. Pendleton, Rings with noetherian spectrum, Duke Math. J.
35 (1968), 631–639. MR 0229627
(37 #5201)
 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
(58 #22034)
 1.
 P. Eakin and A. Sathaye, Prestable ideals, J. Algebra 41 (1976), 439454. MR 54:7449
 2.
 S. Gabelli, Complete integrally closed domains and tideals, Bolletino U.M.I (7) (1989), 327342. MR 90h:13019
 3.
 R. Gilmer, On factorization into prime ideals, Commentarii Math. Helvetici 47 (1972), 7074. MR 46:5310
 4.
 R. Gilmer, Multiplicative Ideal Theory, Queen's Papers Pure Appl. Math. Vol 90, Kingston, 1992. MR 37:5198
 5.
 R. Gilmer, Commutative Semigroup Rings, Chicago Lectures in Math., Chicago, 1984. MR 85e:20058
 6.
 W. Heinzer, D. Lantz, and K. Shah, The RatliffRush ideals in a Noetherian ring, Comm. in Algebra 20 (1992), 591622. MR 93c:13002
 7.
 I. Kaplansky, Commutative Rings, Revised Edition, The University of Chicago Press, 1974. MR 49:10674
 8.
 M. Nagata Local Rings , Interscience, New York, 1962. MR 27:5790
 9.
 J. Ohm and R. Pendleton, Rings with Noetherian spectrum, Duke Math. J. 35 (1968), 631639; Addendum, 875. MR 37:5201
 10.
 L. J. Ratliff, Jr., and D. E. Rush, Two notes on reductions of ideals, Indiana Univ. Math. J. 27 (1978), 929934. MR 58:22034
Similar Articles
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 323064510
Email:
gilmer@math.fsu.edu
William Heinzer
Affiliation:
Department of Mathematics, Purdue University, West Lafayette, Indiana 479071395
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:
http://dx.doi.org/10.1090/S0002993999051990
PII:
S 00029939(99)051990
Keywords:
Cohen's theorem,
finite generation,
maximal ideal,
monoid ring,
Noetherian,
power of an ideal,
RatliffRush 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
