On ideals with the Rees property

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.