## 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 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 (X^{s} : 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 M^{2} is 3-generated.

Original language | English |
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Pages (from-to) | 3141-3151 |

Number of pages | 11 |

Journal | Proceedings of the American Mathematical Society |

Volume | 127 |

Issue number | 11 |

DOIs | |

State | Published - 1999 |

## Keywords

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

## ASJC Scopus subject areas

- General Mathematics
- Applied Mathematics