Narrow Results

We study the number of generators of ideals in regular rings and ask the question whether $\mu (I)<\mu (I^2)$ if $I$ is not a principal ideal, where $\mu (J)$ denotes the number of generators of an ideal $J$. We provide lower bounds for the number of generators for the powers of an ideal and also show that the CM-type of $I^2$ is $\ge 3$ if $I$ is a monomial ideal of height $n$ in $K[x_1,\dots ,x_n]$ and $n\ge 3$.

Let $K$ be a field and $R=K[x_1,\dots ,x_n]$ be a polynomial ring in the variables $x_1,\dots ,x_n$. In this paper, we introduce two classes of monomial ideals of $R$, which have the following properties:

• ⁽ⁱ⁾The (strong) persistence property of associated prime ideals.
• ⁽ⁱⁱ⁾There exists a strongly superficial element.
• ⁽ⁱⁱⁱ⁾Ratliff–Rush closed.

Next, we characterize these monomial ideals. In the sequel, we give some combinatorial aspects. We conclude this paper with constructing new monomial ideals, which have the persistence property.

We prove that the depth functions of cover ideals of balanced hypergraph have the nonincreasing property. Furthermore, we also give a bound for the index of depth stability of these ideals.

We prove that if G is a gap-free and chair-free simple graph, then the regularity of the edge ideal of G is no more than 3. If G is a gap-free and P₄-free graph, then it is a chair-free graph; furthermore, the complement of G is chordal, and thus the regularity of G is 2.