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- Title
On ideals with the Rees property.
- Authors
Migliore, Juan; Miró-Roig, Rosa M.; Murai, Satoshi; Nagel, Uwe; Watanabe, Junzo
- Abstract
A homogeneous ideal I of a polynomial ring S is said to have the Rees property if, for any homogeneous ideal $${J \subset S}$$ J ⊂ S which contains I, the number of generators of J is smaller than or equal to that of I. A homogeneous ideal $${I \subset S}$$ I ⊂ S is said to be $${\mathfrak{m}}$$ m -full if $${\mathfrak{m}I:y=I}$$ m I : y = I for some $${y \in \mathfrak{m}}$$ y ∈ m , where $${\mathfrak{m}}$$ m is the graded maximal ideal of $${S}$$ S . It was proved by one of the authors that $${\mathfrak{m}}$$ 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 $${\mathfrak{m}}$$ 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.
- Publication
Archiv der Mathematik, 2013, Vol 101, Issue 5, p445
- ISSN
0003-889X
- Publication type
Article
- DOI
10.1007/s00013-013-0565-5