## Abstract

Let R be a Noetherian local ring with the maximal ideal m. Assume that R contains ideals I and J satisfying the conditions (1) I ⊆ J, (2) I : m ⊈ J, and (3) J is m-full, that is mJ : x = J for some x ∈ m. Then the theorem says that R is a regular local ring, if the projective dimension pd_{R}I of I is finite. Let q = (a_{1}, a_{2},..., a_{t})R be an ideal in a Noetherian local ring R generated by a maximal R-regular sequence a_{1}, a_{2},..., a_{t} and let q̄ denote the integral closure of q. Then, thanks to the theorem applied to the ideals I = q : m and J = q̄, it follows that I^{2} = qI, unless A is a regular local ring. Consequences are discussed.

Original language | English |
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Pages (from-to) | 631-639 |

Number of pages | 9 |

Journal | Kyoto Journal of Mathematics |

Volume | 42 |

Issue number | 4 |

DOIs | |

Publication status | Published - Feb 2003 |

Externally published | Yes |

## Keywords

- Associated graded ring
- Cohen-Macaulay local ring
- Gorenstein local ring
- Integrally closed ideal
- Projective dimension
- Rees algebra
- Regular local ring
- m-full ideal

## ASJC Scopus subject areas

- Mathematics(all)