On ideals with the Rees property

摘要

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.