Related papers: Unmixed polymatroidal ideals

Let S be a polynomial ring in n variables, over an arbitrary field. We give the total, graded, and multigraded Betti numbers of S/M, for every monomial ideal M in S. We also give an explicit characterization of all monomial ideals M in S…

The class of equidimensional polymatroidal ideals are studied. In particular, we show that an unmixed polymatroidal ideal is connected in codimension one if and only if it is Cohen-Macaulay. Especially a matroidal ideal is connected in…

We study ideal-theoretic conditions for a monomial ideal to be Golod. For ideals in a polynomial ring in three variables, our criteria give a complete characterization. Over such rings, we show that the product of two monomial ideals is…

Let $K$ be a field of characteristic zero, let $I \subset S = K[x_1,\dots,x_n]$ be a homogeneous ideal, and let $\partial(I)$ be its gradient ideal. We study the relationship between $\mathrm{reg}\,I$ and $\mathrm{reg}\,\partial(I)$. While…

We characterize monomial ideals which are intersections of monomial prime ideals and study classes of ideals with this property, among them polymatroidal ideals.

This paper investigates atomic factorizations in the monoid $\mathcal I(R)$ of nonzero ideals of a multivariate polynomial ring $R$, under ideal multiplication. Building on recent advances in factorization theory for unit-cancellative…

The aim of this paper is to unveil an unexpected relationship between the normal form of a polynomial with respect to a polynomial ideal and the more geometric concept of orthogonality. We present a new way to calculate the normal form of a…

Let $R=k[x_1,..., x_n]$ be a polynomial ring over a field $k$ of characteristic $p>0,$ and let $I=(f_1,...,f_s)$ be an ideal of $R.$ We prove that every associated prime $P$ of $H^i_I(R)$ satisfies $\text{dim}R/P\geqslant…

The goal of this paper is to present examples of families of homogeneous ideals in the polynomial ring over a field that satisfy the following condition: every product of ideals of the family has a linear free resolution. As we will see,…

In this paper, we establish some criteria to detect the presence of the maximal ideal $(x_1, \ldots, x_n)$ in the set of associated primes of powers of monomial ideals in the polynomial ring $K[x_1, \ldots, x_n]$. Furthermore, for each of…

An ideal $I$ in a Noetherian ring is called \textit{normal} if $I^n$ is integrally closed for all $n \geq 1$. Zariski proved that in two-dimensional regular local rings, every integrally closed ideal is normal. However, in dimension three…

Given a $d \times n$ integer matrix $A$, the main result is an elementary, simple-to-state algorithm that finds the largest $A$-graded ideal contained in any ideal $I$ in a polynomial ring $\Bbbk[x_1,\ldots,x_n]$. The special case where $A$…

Let R be a nil ring. We prove that primitive ideals in the polynomial ring R[x] in one indeterminate over R are of the form I[x] for some ideals I of R.

In a commutative ring $R$ with unity, given an ideal $I$ of $R$, Anderson and Badawi in 2011 introduced the invariant $\omega(I)$, which is the minimal integer $n$ for which $I$ is an $n$-absorbing ideal of $R$. In the specific case that $R…

Let $R=K[x_1,...,x_n]$ be the polynomial ring in $n$ variables over a field $K$ with the maximal ideal $\frak{m}=(x_1,...,x_n)$. Let $\astab(I)$ and $\dstab(I)$ be the smallest integer $n$ for which $\Ass(I^n)$ and $\depth(I^n)$ stabilize,…

The main goal of the paper is the discussion of a deeper interaction between matrix theory over polynomial rings over a field and typical methods of commutative algebra and related algebraic geometry. This is intended in the sense of…

In this paper we study prime and maximal ideals in a Hurwitz polynomial ring hR. It is well-known that to study many questions we may assume R is prime and consider just R-disjoint ideals. We give a characterization for an R-disjoint ideal…

Let $(R,\mathfrak{m})$ be a quasi-unmixed local ring and $I$ an equimultiple ideal of $R$ of analytic spread $s$. In this paper, we introduce the equimultiple coefficient ideals. Fix $k\in \{1,...,s\}.$ The largest ideal $L$ containing $I$…

For a monomial ideal $I$, we consider the $i$th homological shift ideal of $I$, denoted by $\text{HS}_i(I)$, that is, the ideal generated by the $i$th multigraded shifts of $I$. Some algebraic properties of this ideal are studied. It is…

Let $S=K[x_1,\ldots,x_n]$ be the ring of polynomials over a field $K$. Given two monomial ideals $0\subset I\subsetneq J \subset S$, we present a new method to compute the Hilbert depth of $J/I$. As an application, we show that if $u\in S$…