Narrow Results

The aim of this work is to compare symbolic and ordinary powers of monomial ideals using commutative algebra and combinatorics. Monomial ideals whose symbolic and ordinary powers coincide are called Simis ideals. Weighted monomial ideals are defined by assigning linear weights to monomials. We examine Simis and normally torsion-free ideals, relate some of the properties of monomial ideals and weighted monomial ideals, and present a structure theorem for edge ideals of $d$-uniform clutters whose ideal of covers is Simis in degree $d$. One of our main results is a combinatorial classification of when the dual of the edge ideal of a weighted oriented graph is Simis in degree $2$.

An Egyptian fraction is a finite sum of distinct rational numbers of the form $\frac{1}{m}$, where $m$ is a nonzero integer. It is well known that every rational number can be expressed as an Egyptian fraction. The purpose of this paper is to explore natural analogs of this concept for commutative integral domains.

Normalization is a fundamental ring-theoretic operation; geometrically it resolves singularities in codimension one. Existing algorithmic methods for computing the normalization rely on a common recipe: successively enlarge the given ring in form of an endomorphism ring of a certain (fractional) ideal until the process becomes stationary. While Vasconcelos' method uses the dual Jacobian ideal, Grauert–Remmert-type algorithms rely on so-called test ideals. For algebraic varieties, one can apply such normalization algorithms globally, locally, or formal analytically at all points of the variety. In this paper, we relate the number of iterations for global Grauert–Remmert-type normalization algorithms to that of its local descendants. We complement our results by a study of ADE singularities. All intermediate singularities occurring in the normalization process are determined explicitly. Besides ADE singularities the process yields simple space curve singularities from the list of Frühbis-Krüger.

This paper studies the integral and complete integral closures of an ideal in an integral domain. By definition, the integral closure of an ideal I of a domain R is the ideal given by I′ ≔ {x ∈ R | x satisfies an equation of the form x^r + a₁x^r-1 + ⋯ + a_r = 0, where a_i ∈ Iⁱfor each i ∈ {1, …, r}}, and the complete integral closure of I is the ideal Ī ≔ {x ∈ R | there exists 0 ≠ = c ∈ R such that cxⁿ ∈ Iⁿfor all n ≥ 1}. An ideal I is said to be integrally closed or complete (respectively, completely integrally closed) if I = I′ (respectively, I = Ī). We investigate the integral and complete integral closures of ideals in many different classes of integral domains and we give a new characterization of almost Dedekind domains via the complete integral closure of ideals.

Let R ⊂ S be an extension of integral domains. We say that (R, S) is a well-centered pair of rings, if each intermediate ring T between R and S is well-centered on R, in the sense that each principal ideal of T is generated by an element of R. The aim of this paper is to study well-centered pairs of rings. We investigate the structure of the intermediate rings T between R and S that are well-centered on R. We establish the relationship between well-centered pairs and normal pairs. We present some examples and counterexamples illustrating our theory and showing the limits of our results.

Let $(R,𝔪)$ be a commutative Noetherian complete local ring and $𝔞$ and $𝔟$ ideals of $R$. Motivated by a question of Rees, we study the relationship between $\overline{𝔟}$, the classical Northcott–Rees integral closure of $𝔟$, and ${𝔟}^{\ast (H)}$, the integral closure of $𝔟$ relative to an Artinian $R$-module $H$ (also called here ST-closure of $𝔟$ on $H$), in order to study a relation between $e(𝔞)$, the multiplicity of $𝔞$, and $e^{\prime }(𝔞;H)$, the multiplicity of $𝔞$ relative to an Artinian $R$-module $H$. We conclude $\overline{𝔟}={𝔟}^{\ast (H)}$ when every minimal prime ideal of $R$ belongs to the set of attached primes of $H$. As an application, we show what happens when $H$ is a generalized local cohomology module.

We provide a characterization of the Arf property in both the numerical duplication of a numerical semigroup and in a member of a family of quotients of the Rees algebra studied in [V. Barucci, M. D’Anna and F. Strazzanti, A family of quotients of the Rees algebra, Commun. Algebra 43(1) (2015) 130–142].

In this paper, the notions of integral closure of hyperrings and hyperideals in a Krasner hyperring $(R,+,\cdot )$ are defined and some basics properties of them are studied. We define also the notion of hypervaluation hyperideals and then a relations between hypervaluations, integral closure of hyperideals and primary hyperideals are studied. In fact it is shown that the integral closure of a hyperideal is determined by the hypervaluation Krasner hyperrings.

We show that the graded structure of an ideal is preserved under root closures. As an application, for a locally nilpotent derivation $\delta$ on a Noetherian integral domain $R$ containing a field of characteristic zero, we show that if $I$ is $\delta$-invariant, then so are its root closures.

This is an exposition of some new results on associated primes and the depth of different kinds of powers of monomial ideals in order to show a deep connection between commutative algebra and some objects in combinatorics such as simplicial complexes, integral points in polyhedrons and graphs.