On ideals with the Rees property

TL;DR

This paper explores ideals with the Rees property, providing examples beyond the known class of m-full ideals, and investigates the Sperner property in Artinian monomial almost complete intersections in three variables.

Contribution

It demonstrates that in polynomial rings with more than two variables, there exist ideals with the Rees property that are not m-full, expanding understanding of ideal properties.

Findings

Existence of Rees property ideals that are not m-full in rings with >2 variables

Artinian monomial almost complete intersections in three variables have the Sperner property

Counterexamples to the equivalence of Rees property and m-fullness in higher variables

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 $\mathfrak mI:y=I$ 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.