Related papers: Interpolation of Ideals

Let $I$ be a monomial ideal in a polynomial ring $S=K[x_1,\ldots,x_n]$ over a field $K$ with $n=2$ or $3$, and let $\overline{I}$ be its integral closure. We will show that $\text{reg} (\overline{I}) \le \text{reg} (I)$. Furthermore, if $I$…

Our focus in this paper is in effective computation of the core core(I) of an ideal I which is defined to be the intersection of all minimal reductions of I. The first main result is a closed formula for the graded core(m) of the maximal…

Let $k$ be an infinite field and $I\subset k [x_1, \ldots ,x_n]$ be an ideal such that dim $V(I)=q$. Denote by $(f_1, \ldots, f_s)$ a set of generators of $I$. One can see that in the set $I\cap k [x_{1},...,x_{q+1}]$ there exist non-zero…

We propose an effective method for primary decomposition of symmetric ideals. Let $K[X]=K[x_1,\ldots,x_n]$ be the $n$-valuables polynomial ring over a field $K$ and $\mathfrak{S}_n$ the symmetric group of order $n$. We consider the…

If I is an ideal in a Gorenstein ring S and S/I is Cohen-Macaulay, then the same is true for any linked ideal I'. However, such statements hold for residual intersections of higher codimension only under very restrictive hypotheses, not…

We consider ideals $I$ in a Stanley-Reisner ring $k[\Delta]$ over the simplical complex $\Delta$, such that the tight closure of $I$, $I^*$, is equal to $\mathfrak{m}$, the standard graded maximal ideal of $k[\Delta]$. We determine the…

Let $I$ be an intersection of three monomial prime ideals of a polynomial algebra $S$ over a field. We give a special Stanley decomposition of $I$ which provides a lower bound of the Stanley depth of $I$, greater than or equal to $\depth\…

D. Rees and J. Sally defined the core of an $R$-ideal $I$ as the intersection of all $($minimal$)$ reductions of $I$. However, it is not easy to give an explicit characterization of it in terms of data attached to the ideal. Until recently,…

Among the several types of closures of an ideal $I$ that have been defined and studied in the past decades, the integral closure $\bar{I}$ has a central place being one of the earliest and most relevant. Despite this role, it is often a…

Given a polynomial ring $C$ over a field and proper ideals $I$ and $J$ whose generating sets involve disjoint variables, we determine how to embed the associated primes of each power of $I+J$ into a collection of primes described in terms…

A minor is principal means it is defined by the same row and column indices. Let $X$ be a square generic matrix, $K[X]$ the polynomial ring in entries of $X$, over an algebraically closed field, $K$. For fixed $t\leq n$, let $\mathfrak P_t$…

Let $\mathbb{K}$ be a field and $R = \mathbb{K}[x_1, \ldots, x_n]$. We obtain an improved upper bound for asymptotic resurgence of squarefree monomial ideals in $R$. We study the effect on the resurgence when sum, product and intersection…

Let I\subset K[x,y] be a <x,y>-primary monomial ideal where K is a field. This paper produces an algorithm for computing the Ratliff-Rush closure I for the ideal I=<m_0,...,m_{n}> whenever m_{i} is contained in the integral closure of the…

The elements of a finite partial order $P$ can be identified with the maximal indecomposable two-sided ideals of its incidence algebra $\A$, and then for two such ideals, $I\prec J \iff IJ \not=0$. This offers one way to recover a poset…

The main aim of this paper is to characterize ideals I in the power series ring R=K[[x1,...,xs]] that are finitely determined up to contact equivalence by proving that this is the case if and only if I is an isolated complete intersection…

When studying local properties of a polynomial ideal, one usually needs a theoretic technique called localization. For most cases, in spite of its importance, the computation in a localized ring cannot be algorithmically preformed. On the…

The main focus of this paper is on the problem of relating an ideal $I$ in the polynomial ring $\mathbb Q[x_1, \dots, x_n]$ to a corresponding ideal in $\mathbb F_p[x_1,\dots, x_n]$ where $p$ is a prime number; in other words, the…

Consider a complete intersection I of type (d_1,..., d_r) in a polynomial ring over a field of characteristic 0. We study the graded system of ideals {gin(I^n)}_n obtained by taking the reverse lexicographic generic initial ideals of the…

The closed one-sided ideals of a C*-algebra are exactly the closed subspaces supported by the orthogonal complement of a closed projection. Let A be a (not necessarily selfadjoint) subalgebra of a unital C*-algebra B which contains the unit…

Let $R$ be a polynomial ring in $N$ variables over an arbitrary field $K$ and let $I$ be an ideal of $R$ generated by $n$ polynomials of degree at most 2. We show that there is a bound on the projective dimension of $R/I$ that depends only…