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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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
Published electronically: May 4, 1999
MathSciNet review: 1646305
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Abstract: Suppose $M$ is a maximal ideal of a commutative integral domain $R$ and that some power $M^n$ of $M$ is finitely generated. We show that $M$ is finitely generated in each of the following cases: (i) $M$ is of height one, (ii) $R$ is integrally closed and $\operatorname{ht} M=2$, (iii) $R = K[X;\tilde S]$ is a monoid domain over a field $K$, where $\tilde S = S \cup \{0\}$ is a cancellative torsion-free monoid such that $\bigcap _{m=1}^\infty mS=\emptyset$, and $M$ is the maximal ideal $(X^s:s\in S)$. We extend the above results to ideals $I$ of a reduced ring $R$ such that $R/I$ is Noetherian. We prove that a reduced ring $R$ is Noetherian if each prime ideal of $R$ has a power that is finitely generated. For each $d$ with $3 \le d \le \infty$, we establish existence of a $d$-dimensional integral domain having a nonfinitely generated maximal ideal $M$ of height $d$ such that $M^2$ is $3$-generated.


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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: http://dx.doi.org/10.1090/S0002-9939-99-05199-0
PII: S 0002-9939(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