Uniformly primary ideals

This article introduces and advances the basic theory of “uniformly primary ideals” for commutative rings, a concept that imposes a certain boundedness condition on the usual notion of “primary ideal”. Characterizations of uniformly primary ideals are provided along with examples that give the theory independent value. Applications are also provided in contexts that are relevant to Noetherian rings.     

Principal Borel ideals and Gotzmann ideals

In this paper we characterize all principal Borel ideals with Borel generator up to degree 4 which are Gotzmann. We also classify principal Borel ideals with a Borel generator of degree d which are lexsegment and we describe the shadows of principal Borel ideals. Finally, we discuss the corresponding results for squarefree monomial ideals.Received: 10 May 2002     

Cohen-Macaulay monomial ideals with given radical

We study the set of Cohen-Macaulay monomial ideals with a given radical. Among this set of ideals are the so-called Cohen-Macaulay modifications. Not all Cohen-Macaulay squarefree monomial ideals admit nontrivial Cohen-Macaulay modifications. It is shown that if there exists one such modification, then there exist indeed infinitely many.     

Algebraic properties of classes of path ideals of posets

We consider path ideals associated to special classes of posets such as tree posets and cycles. We express their property of being sequentially Cohen–Macaulay in terms of the underlying poset. Moreover, monomial ideals, which arise in algebraic statistics from the Luce-decomposable model and the ascending model, can be viewed as path ideals of certain posets. We study invariants of these so-called Luce-decomposable monomial ideals and ascending ideals for diamond posets and products of chains. In particular, for these classes of posets, we explicitly compute their Krull dimension, their projective dimension, their Castelnuovo–Mumford regularity and their Betti numbers.     

On Macaulay duals of Hilbert ideals

In this note we present some computational tools to aide in the determination of Macaulay inverses of Hilbert ideals of finite groups and related ideals.     

Gröbner bases of ideals cogenerated by Pfaffians

We characterise the class of one-cogenerated Pfaffian ideals whose natural generators form a Gröbner basis with respect to any anti-diagonal term order. We describe their initial ideals as well as the associated simplicial complexes, which turn out to be shellable and thus Cohen-Macaulay. We also provide a formula for computing their multiplicity.     

Strongly irreducible ideals of a commutative ring

An ideal I of a ring R is said to be strongly irreducible if for ideals J and K of R, the inclusion J∩K⊆I implies that either J⊆I or K⊆I. The relationship among the families of irreducible ideals, strongly irreducible ideals, and prime ideals of a commutative ring R is considered, and a characterization is given of the Noetherian rings which contain a non-prime strongly irreducible ideal.     

Independence numbers of graphs and generators of ideals

This article investigates the generators of certain homogeneous ideals which are associated with graphs with bounded independence numbers. These ideals first appeared in the theory oft-designs. The main theorem suggests a new approach to the Clique Problem which isNP-complete. This theorem has a more general form in commutative algebra dealing with ideals associated with unions of linear varieties. This general theorem is stated in the article; a corollary to it generalizes Turán’s theorem on the maximum graphs with a prescribed clique number. Research supported in part by NSF Grant MCS77-03533.     

Growth conditions for a family of ideals containing regular sequences

It has been conjectured by Eisenbud-Green-Harris that lex-plus-powers ideals exhibit extremal conditions among all homogeneous ideals containing a regular sequence of forms in fixed degrees. In the same spirit, we consider a family of homogeneous ideals in k[x,y,z] which contain a regular sequence of forms F,G∈k[x,y] and compare the growth of these ideals with special monomial ideals sharing similar properties.     

Generalized binomial edge ideals

This paper studies a class of binomial ideals associated to graphs with finite vertex sets. They generalize the binomial edge ideals, and they arise in the study of conditional independence ideals. A Gröbner basis can be computed by studying paths in the graph. Since these Gröbner bases are square-free, generalized binomial edge ideals are radical. To find the primary decomposition a combinatorial problem involving the connected components of subgraphs has to be solved. The irreducible components of the solution variety are all rational.     

On the integral closure of ideals

Among the several types of closures of an ideal I that have been defined and studied in the past decades, the integral closure has a central place being one of the earliest and most relevant. Despite this role, it is often a difficult challenge to describe it concretely once the generators of I are known. Our aim in this note is to show that in a broad class of ideals their radicals play a fundamental role in testing for integral closedness, and in case , is still helpful in finding some fresh new elements in . Among the classes of ideals under consideration are: complete intersection ideals of codimension two, generic complete intersection ideals, and generically Gorenstein ideals. Received: 28 July 1997     

On the integral closure of ideals

Among the several types of closures of an idealI that have been defined and studied in the past decades, the integral closureĪ has a central place being one of the earliest and most relevant. Despite this role, it is often a difficult challenge to describe it concretely once the generators ofI are known. Our aim in this note is to show that in a broad class of ideals their radicals play a fundamental role in testing for integral closedness, and in caseI ≠Ī, ✓I is still helpful in finding some fresh new elements inĪ/I. Among the classes of ideals under consideration are: complete intersection ideals of codimension two, generic complete intersection ideals, and generically Gorenstein ideals. Part of the results contained in this paper were obtained while the first author was visiting Rutgers University and was partially supported by CNR grant 203.01.63, Italy. The second and third authors were partially supported by the NSF. This article was processed by the author using the LAT_EX style filecljour1 from Springer-Verlag     

Ideals containing the squares of the variables

We study the Betti numbers of graded ideals containing the squares of the variables, in a polynomial ring. We prove the lex-plus-powers conjecture for such ideals.     

Tameness of local cohomology of monomial ideals with respect to monomial prime ideals

In this paper we consider the local cohomology of monomial ideals with respect to monomial prime ideals and show that all these local cohomology modules are tame.     

Codimension and projective dimension up to symmetry

Symmetric ideals in increasingly larger polynomial rings that form an ascending chain are investigated. We focus on the asymptotic behavior of codimensions and projective dimensions of ideals in such a chain. If the ideals are graded it is known that the codimensions grow eventually linearly. Here this result is extended to chains of arbitrary symmetric ideals. Moreover, the slope of the linear function is explicitly determined. We conjecture that the projective dimensions also grow eventually linearly. As part of the evidence we establish two non-trivial lower linear bounds of the projective dimensions for chains of monomial ideals. As an application, this yields Cohen–Macaulayness obstructions.     

On the Homology and Fiber Cone of Ideals

In this article, we give a unified approach for several results concerning the fiber cone. Our novel idea is to use the complex C(x _k , ? _{I 1; I 2} , (1, n)). We improve earlier results obtained by several researchers and get some new results. We give a more general definition of ideals of minimal multiplicity and of ideals of almost minimal multiplicity. We also compute the Hilbert series of the fiber cone for these ideals.     

A remark on regularity of powers and products of ideals

We give a simple proof for the fact that the Castelnuovo–Mumford regularity and related invariants of products of powers of ideals are asymptotically linear in the exponents, provided that each ideal is generated by elements of constant degree. We provide examples showing that the asymptotic linearity is false in general. On the other hand, the regularity is always given by the maximum of finitely many linear functions whose coefficients belong to the set of the degrees of generators of the ideals.     

Some results relating to adjacent ideals in dimension two

We answer some of the questions posed by Noh in [S. Noh, Adjacent integrally closed ideals in dimension two, J. Pure Appl. Algebra 85 (2) (1993) 163-184] concerning the existence of adjacent complete ideals in dimension two.