Weak subintegral closure of ideals

We describe some basic facts about the weak subintegral closure of ideals in both the algebraic and complex-analytic settings. We focus on the analogy between results on the integral closure of ideals and modules and the weak subintegral closure of an ideal. We start by giving a new geometric interpretation of the Reid–Roberts–Singh criterion for when an element is weakly subintegral over a subring. We give new characterizations of the weak subintegral closure of an ideal. We associate with an ideal I of a ring A an ideal I_>, which consists of all elements of A such that v(a)>v(I), for all Rees valuations v of I. The ideal I_> plays an important role in conditions from stratification theory such as Whitney's condition A and Thom's condition Af and is contained in every reduction of I. We close with a valuative criterion for when an element is in the weak subintegral closure of an ideal. For this, we introduce a new closure operation for a pair of modules, which we call relative weak closure. We illustrate the usefulness of our valuative criterion.     

Note on some ideals of associative rings

It is well known that being an ideal (left ideal, right ideal) of a ring is not a transitive relation. Nevertheless in some cases the transitive property does hold. Systematic studies of this subject were started in [11,13]. In this paper we continue these studies.     

Integral degree of a ring,reduction numbers and uniform Artin–Rees numbers

The supremum of reduction numbers of ideals having principal reductions is expressed in terms of the integral degree, a new invariant of the ring, which is finite provided the ring has finite integral closure. As a consequence, one obtains bounds for the Castelnuovo–Mumford regularity of the Rees algebra and for the Artin–Rees numbers.     

On essential extensions of reduced rings and domains

A ring is said to be a left essential extension of a reduced ring (domain) if it contains a left ideal which is a reduced ring (domain) and intersects nontrivially every nonzero twosided ideal of the ring. We prove that every ring which is a left essential extension of a reduced ring is a subdirect sum of rings which are essential extensions of domains, but the converse implication does not hold. We give some applications of this result and discuss several related questions.Received: 6 January 2003     

The Extensional Structure of Commutative Noetherian Rings

Abstract

We show that finitely generated modules over a commutative Noetherian ring can be classified, up to isomorphism of submodule series, in a manner analogous to the classification of integers as products of prime numbers. In outline, two such modules have isomorphic submodule series if and only if 1) the set of minimal associated prime ideals of these modules coincide, 2) the multiplicities of these modules at these prime ideals coincide, and 3) the modules represent the same element in a certain group corresponding to the above set of prime ideals. Regarding condition 3), we show that, in the very special case that the ring is a Dedekind domain, the group corresponding to the prime ideal (0) is the ideal class group of the ring.     

Separable integral extensions and plus closure

We show that an excellent local domain of characteristic p has a separable big Cohen–Macaulay algebra. In the course of our work we prove that an element which is in the Frobenius closure of an ideal can be forced into the expansion of the ideal to a module-finite separable extension ring. Received: 28 May 1998 / Revised version: 30 November 1998     

Associated Primes of the Example of Brenner and Monsky

Recently, Brenner and Monsky found an example of an ideal in a hypersurface ring whose tight closure does not commute with localization, thus answered the localization problem in tight closure theory in the negative. In this article, we use Monsky's calculations to analyze the set of associated primes of the Frobenius powers of this ideal and show that this set is infinite.     

On strongly Π-regular ideals

In this paper, we introduce the concept of strongly π-regular ideal of a ring. We prove that every square regular matrix over a strongly π-regular ideal of a ring admits a diagonal reduction.     

Structure of Gröbner bases with respect to block orders

In this paper we study the structure of Gröbner bases with respect to block orders. We extend Lazard's theorem and the Gianni-Kalkbrenner theorem to the case of a zero-dimensional ideal whose trace in the ring generated by the first block of variables is radical. We then show that they do not hold for general zero-dimensional ideals.

     

Looking out for stable syzygy bundles

We study (slope-)stability properties of syzygy bundles on a projective space PN given by ideal generators of a homogeneous primary ideal. In particular we give a combinatorial criterion for a monomial ideal to have a semistable syzygy bundle. Restriction theorems for semistable bundles yield the same stability results on the generic complete intersection curve. From this we deduce a numerical formula for the tight closure of an ideal generated by monomials or by generic homogeneous elements in a generic two-dimensional complete intersection ring.     

The ascending chain condition for real ideas

If A is a ring satisfying the ascending chain condition for real ideals, then this condition is also satisfied by the polynomial ring A[X]. However, an example is given to illustrate that the condition need not to hold in the power series ring A[[X]]. It is also shown that if every real prime ideal is the real ideal generated by finitely many elements, then the ring satisfies the ascending chain condition for real ideals. So, the analogues of Hilbert basis theorem and Cohen's theorem hold for real ideals.     

Hilbert schemes and maximal Betti numbers over veronese rings

Macaulay??s Theorem (Macaulay in Proc. Lond Math Soc 26:531?C555, 1927) characterizes the Hilbert functions of graded ideals in a polynomial ring over a field. We characterize the Hilbert functions of graded ideals in a Veronese ring R (the coordinate ring of a Veronese embedding of P ^r-1). We also prove that the Hilbert scheme, which parametrizes all graded ideals in R with a fixed Hilbert function, is connected; this is an analogue of Hartshorne??s Theorem (Hartshorne in Math. IHES 29:5?C48, 1966) that Hilbert schemes over a polynomial ring are connected. Furthermore, we prove that each lex ideal in R has the greatest Betti numbers among all graded ideals with the same Hilbert function.     

Height bounds,nullstellensatz and primality

In this study, we find height bounds in the polynomial ring over the field of algebraic numbers to test the primality of an ideal. We also obtain height bounds in the arithmetic Nullstellensatz. We apply nonstandard analysis and hence our constants will be ineffective.     

Tightly closed ideals of small type

We study the smallest possible type of tightly closed ideals that are cofinal with the powers of the maximal ideal; this numerical invariant yields information about the tight closure of arbitrary ideals in the ring.

     

Coloring of Plane Graphs with Unique Maximal Colors on Faces

The Four Color Theorem asserts that the vertices of every plane graph can be properly colored with four colors. Fabrici and Göring conjectured the following stronger statement to also hold: the vertices of every plane graph can be properly colored with the numbers 1, …, 4 in such a way that every face contains a unique vertex colored with the maximal color appearing on that face. They proved that every plane graph has such a coloring with the numbers 1, …, 6. We prove that every plane graph has such a coloring with the numbers 1, …, 5 and we also prove the list variant of the statement for lists of sizes seven.     

Homological properties of Orlik–Solomon algebras

The Orlik–Solomon algebra of a matroid can be considered as a quotient ring over the exterior algebra E. At first, we study homological properties of E-modules as e.g., complexity, depth and regularity. In particular, we consider modules with linear injective resolutions. We apply our results to Orlik–Solomon algebras of matroids and give formulas for the complexity, depth and regularity of such rings in terms of invariants of the matroid. Moreover, we characterize those matroids whose Orlik–Solomon ideal has a linear projective resolution and compute in these cases the Betti numbers of the ideal.     

Colength,multiplicity, and ideal closure operations

Abstract

In a formally unmixed Noetherian local ring, if the colength and multiplicity of an integrally closed ideal agree, then R is regular. We deduce this using the relationship between multiplicity and various ideal closure operations.     

PURE SUBMODULES AND TORSION FREE IMAGES OF COMPLETELY DECOMPOSABLE TORSION FREE MODULES

ABSTRACT

Prebalanced and precobalanced sequences play an important role in the investigation of Butler Modules. For Butler groups (modules over the integers), they are equivalent conditions. This is not the case for modules over integral domains in general. We investigate conditions when one type of exactness would imply the other. We show that for analytically unramified domains, the equivalence of prebalanced and precobalanced exactness will hold if and only if every maximal ideal has a unique maximal ideal lying over it in the domain's integral closure.     

Growth of primary decompositions of Frobenius powers of ideals

It was previously known, by work of Smith–Swanson and of Sharp–Nossem, that the linear growth property of primary decompositions of Frobenius powers of ideals in rings of prime characteristic has strong connections to the localization problem in tight closure theory. The localization problem has recently been settled in negative by Brenner and Monsky, but the linear growth question is still open. We study growth of primary decompositions of Frobenius powers of dimension one homogeneous ideals in graded rings over fields. If the ring is positively graded we prove that the linear growth property holds. For non-negatively graded rings we are able to show that there is a “polynomial growth”. We present explicit primary decompositions of Frobenius powers of an ideal, which were known to have infinitely many associated primes, having this linear growth property. We also discuss some other interesting examples.     

On Almost Valuation and Almost Bézout Rings

In this paper we extend the notion of almost valuation and almost Bézout domains to arbitrary commutative rings, and we investigate the transfer of these properties to trivial ring extensions and amalgamated algebras along an ideal. Our aim is to provide new classes of commutative rings satisfying these properties. As an immediate consequence, we show the failure of Anderson–Zafrullah's theorem on the integral closure of an almost valuation domain beyond the context of integral domains.