Finite generation of powers of ideals

Robert Gilmer, William Heinzer, Moshe Roitman

Research output: Contribution to journalArticlepeer-review


Suppose M is a maximal ideal of a commutative integral domain R and that some power Mn 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 ht M = 2, (iii) R = K[X;S̃] is a monoid domain over a field K, where S̃ = S ∪ {0} is a cancellative torsion-free monoid such that ∩m=1 mS = 0, and M is the maximal ideal (Xs : s ε 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 ≤ d ≤ ∞, we establish existence of a d-dimcnsional integral domain having a nonfinitely generated maximal ideal M of height d such that M2 is 3-generated.

Original languageEnglish
Pages (from-to)3141-3151
Number of pages11
JournalProceedings of the American Mathematical Society
Issue number11
StatePublished - 1999


  • Cohen's theorem
  • Finite generation
  • Maximal ideal
  • Monoid ring
  • Noetherian
  • Power of an ideal
  • Ratliff-Rush closure

ASJC Scopus subject areas

  • Mathematics (all)
  • Applied Mathematics


Dive into the research topics of 'Finite generation of powers of ideals'. Together they form a unique fingerprint.

Cite this