On ideals with the Rees property

Abstract

A homogeneous ideal $I$ of a polynomial ring $S$ is said to have the Rees property if, for any homogeneous ideal $J\subset S$ which contains $I$, the number of generators of $J$ is smaller than or equal to that of $I$. A homogeneous ideal $I\subset S$ is said to be $\mathfrak{m}$-full if $\mathfrak{m}I:y=I$ for some $y\in \mathfrak{m}$, where $\mathfrak{m}$ is the graded maximal ideal of $S$. It was proved by one of the authors that $\mathfrak{m}$-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 $\mathfrak{m}$-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.

8 pages