### Abstract

A homogeneous ideal I of a polynomial ring S is said to have the Rees property if, for any homogeneous ideal J ⊂ S which contains I, the number of generators of J is smaller than or equal to that of I. A homogeneous ideal I ⊂ S is said to be m-full if mI:y=I for some y ∈ m, where m is the graded maximal ideal of S. It was proved by one of the authors that m-full ideals have the Rees property and that the converse holds in a polynomial ring with two variables. In this note, we give examples of ideals which have the Rees property but are not m-full in a polynomial ring with more than two variables. To prove this result, we also show that every Artinian monomial almost complete intersection in three variables has the Sperner property.

Original language | English |
---|---|

Pages (from-to) | 445-454 |

Number of pages | 10 |

Journal | Archiv der Mathematik |

Volume | 101 |

Issue number | 5 |

DOIs | |

Publication status | Published - 2013 Nov 1 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Archiv der Mathematik*,

*101*(5), 445-454. https://doi.org/10.1007/s00013-013-0565-5

**On ideals with the Rees property.** / Migliore, Juan; Miró-Roig, Rosa M.; Murai, Satoshi; Nagel, Uwe; Watanabe, Junzo.

Research output: Contribution to journal › Article

*Archiv der Mathematik*, vol. 101, no. 5, pp. 445-454. https://doi.org/10.1007/s00013-013-0565-5

}

TY - JOUR

T1 - On ideals with the Rees property

AU - Migliore, Juan

AU - Miró-Roig, Rosa M.

AU - Murai, Satoshi

AU - Nagel, Uwe

AU - Watanabe, Junzo

PY - 2013/11/1

Y1 - 2013/11/1

N2 - A homogeneous ideal I of a polynomial ring S is said to have the Rees property if, for any homogeneous ideal J ⊂ S which contains I, the number of generators of J is smaller than or equal to that of I. A homogeneous ideal I ⊂ S is said to be m-full if mI:y=I for some y ∈ m, where m is the graded maximal ideal of S. It was proved by one of the authors that m-full ideals have the Rees property and that the converse holds in a polynomial ring with two variables. In this note, we give examples of ideals which have the Rees property but are not m-full in a polynomial ring with more than two variables. To prove this result, we also show that every Artinian monomial almost complete intersection in three variables has the Sperner property.

AB - A homogeneous ideal I of a polynomial ring S is said to have the Rees property if, for any homogeneous ideal J ⊂ S which contains I, the number of generators of J is smaller than or equal to that of I. A homogeneous ideal I ⊂ S is said to be m-full if mI:y=I for some y ∈ m, where m is the graded maximal ideal of S. It was proved by one of the authors that m-full ideals have the Rees property and that the converse holds in a polynomial ring with two variables. In this note, we give examples of ideals which have the Rees property but are not m-full in a polynomial ring with more than two variables. To prove this result, we also show that every Artinian monomial almost complete intersection in three variables has the Sperner property.

UR - http://www.scopus.com/inward/record.url?scp=84887477508&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84887477508&partnerID=8YFLogxK

U2 - 10.1007/s00013-013-0565-5

DO - 10.1007/s00013-013-0565-5

M3 - Article

AN - SCOPUS:84887477508

VL - 101

SP - 445

EP - 454

JO - Archiv der Mathematik

JF - Archiv der Mathematik

SN - 0003-889X

IS - 5

ER -