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.
ASJC Scopus subject areas
- 数学 (全般)