Finite generation of powers of ideals
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- by Robert Gilmer, William Heinzer and Moshe Roitman PDF
- Proc. Amer. Math. Soc. 127 (1999), 3141-3151 Request permission
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.References
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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
- 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
- © Copyright 1999 American Mathematical Society
- 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
- MathSciNet review: 1646305