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Finite generation of powers of ideals

Authors:	Robert Gilmer William Heinzer Moshe Roitman

Institution:	Department of Mathematics, Florida State University Tallahassee, Florida 32306-4510 ; Department of Mathematics, Purdue University, West Lafayette, Indiana 47907-1395 ; Department of Mathematics, University of Haifa, Mount Carmel, Haifa 31905, Israel

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 $\operatorname{ht} M=2$ , (iii) $R = KX;\tilde S]$ is a monoid domain over a field , where $\tilde S = S \cup \{0\}$ is a cancellative torsion-free monoid such that $\bigcap _{m=1}^\infty mS=\emptyset$ , and is the maximal ideal $(X^s:s\in S)$ . 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 $3 \le d \le \infty$ , we establish existence of a -dimensional integral domain having a nonfinitely generated maximal ideal of height such that is -generated.

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

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