Integral Closure Research Articles

ANALYTIC SPREAD AND INTEGRAL CLOSURE OF INTEGRALLY DECOMPOSABLE MODULES

AbstractWe relate the analytic spread of a module expressed as the direct sum of two submodules with the analytic spread of its components. We also study a class of submodules whose integral closure can be expressed in terms of the integral closure of its row ideals, and therefore can be obtained by means of a simple computer algebra procedure. In particular, we analyze a class of modules, not necessarily of maximal rank, whose integral closure is determined by the family of Newton polyhedra of their row ideals.

Associated primes and integral closure of Noetherian rings

Let A be a Noetherian ring and B be a finitely generated A-algebra. Denote by A' the integral closure of A in B. We give necessary and sufficient conditions for prime ideals to be in Ass_{A}(B/A') and Ass_{A'}(B/A') generalizing and strengthening classical results for rings of special type.

Multiplicity sequence and integral dependence

We prove that two arbitrary ideals $I \subset J$ in an equidimensional and universally catenary Noetherian local ring have the same integral closure if and only if they have the same multiplicity sequence. We also obtain a Principle of Specialization of Integral Dependence, which gives a condition for integral dependence in terms of the constancy of the multiplicity sequence in families.

Almost discrete valuation domains

Let D be an integral domain. Then D is an almost valuation (AV)-domain if for there exists a natural number n with or AV-domains are closely related to valuation domains, for example, D is an AV-domain if and only if the integral closure is a valuation domain and is a root extension. In this note, we explore various generalizations of DVRs (which we might call almost DVRs) such as Noetherian AV-domains, AV-domains with a DVR, and quasilocal and local API-domains (i.e., for there exists an n with principal). The structure of complete local AV-domains and API-domains is determined.

Integral closures in real algebraic geometry

We study the algebraic and geometric properties of the integral closure of different rings of functions on a real algebraic variety: the regular functions and the continuous rational functions.

On almost valuation ring pairs

If [Formula: see text] are (commutative) rings, [Formula: see text] denotes the set of intermediate rings and [Formula: see text] is called an almost valuation (AV)-ring pair if each element of [Formula: see text] is an AV-ring. Many results on AV-domains and their pairs are generalized to the ring-theoretic setting. Let [Formula: see text] be rings, with [Formula: see text] denoting the integral closure of [Formula: see text] in [Formula: see text]. Then [Formula: see text] is an AV-ring pair if and only if both [Formula: see text] and [Formula: see text] are AV-ring pairs. Characterizations are given for the AV-ring pairs arising from integrally closed (respectively, integral; respectively, minimal) ring extensions [Formula: see text]. If [Formula: see text] is an AV-ring pair, then [Formula: see text] is a P-extension. The AV-ring pairs [Formula: see text] arising from root extensions are studied extensively. Transfer results for the “AV-ring” property are obtained for pullbacks of [Formula: see text] type, with applications to pseudo-valuation domains, integral minimal ring extensions, and integrally closed maximal non-AV subrings. Several sufficient conditions are given for [Formula: see text] being an AV-ring pair to entail that [Formula: see text] is an overring of [Formula: see text], but there exist domain-theoretic counter-examples to such a conclusion in general. If [Formula: see text] is an AV-ring pair and [Formula: see text] satisfies FCP, then each intermediate ring either contains or is contained in [Formula: see text]. While all AV-rings are quasi-local going-down rings, examples in positive characteristic show that an AV-domain need not be a divided domain or a universally going-down domain.

Integral closure of bipartite graph ideals

Combinatorial properties of some ideals related to bipartite graphs are examined. A description of the integral closure expressed through the log set of edge ideals of complete bipartite graphs is illustrated together with the fact that nontrivial generalized graph ideals of a strong complete quasi-bipartite graph are integrally closed.

On finitely stable domains, II

Among other results, we prove the following: A locally Archimedean stable domain satisfies accp. A stable domain R is Archimedean if and only if every nonunit of R belongs to a height-one prime ideal of the integral closure R ′ of R in its quotient field (this result is related to Ohm’s theorem for Prüfer domains). An Archimedean stable domain R is one-dimensional if and only if R ′ is equidimensional (generally, an Archimedean stable local domain is not necessarily one-dimensional). An Archimedean finitely stable semilocal domain with stable maximal ideals is locally Archimedean, but generally, neither Archimedean stable domains, nor Archimedean semilocal domains are necessarily locally Archimedean.

Integral Domains Whose w-Integral Closures Are Krull Domains

Let D be an integral domain with quotient field K, [Formula: see text] be the integral closure of D in K, and D[w] be the w-integral closure of D in K; so [Formula: see text], and equality holds when D is Noetherian or dim(D) = 1. The Mori–Nagata theorem states that if D is Noetherian, then [Formula: see text] is a Krull domain; it has also been investigated when [Formula: see text] is a Dedekind domain. We study integral domains D such that D[w] is a Krull domain. We also provide an example of an integral domain D such that [Formula: see text], t-dim(D) = 1, [Formula: see text] is a Prüfer v-multiplication domain with t-dim([Formula: see text]) = 2, and D[w] is a UFD.

On Ratliff-Rush closure of modules

In this paper, we introduce the notion of Ratliff-Rush closure of modules and explore whether the condition of the Ratliff-Rush closure coincides with the integral closure. The main result characterizes the condition in terms of the normality of the projective scheme of the Rees algebra. In conclusion, we shall give a criterion for the Buchsbaum Rees algebras.

Unique decomposition into ideals for commutative Noetherian rings

A commutative ring R has the unique decomposition into ideals (UDI) property if, for any R-module that decomposes into a finite direct sum of indecomposable ideals, this decomposition is unique up to the order and isomorphism classes of the indecomposable ideals. In a 2001 paper, Goeters and Olberding characterize the UDI property for Noetherian integral domains, noting the almost local nature of the property and, in the case of a local domain, relating it to the structure of its integral closure. In a 2011 paper, Ay and Klingler obtain similar results for Noetherian reduced rings. In this paper, we examine the UDI property for arbitrary commutative Noetherian rings, establishing the same almost local nature of the property, and giving an example which shows that the local results do not extend to commutative Noetherian rings in general.

Exact irreducible moments of the Landau collision operator in the random-velocity moment expansion

Exact moments of the Landau collision operator are calculated for the irreducible Hermite polynomials written in terms of the random-velocity variable. We present closed, algebraic formulas which can be implemented in computer algebra systems. The formulas reproduce the results for the total-velocity moment expansion (J-Y Ji and E D Held, 2006 Phys. Plasmas 13 102 103) and for the random-velocity moment expansion with the small mass-ratio approximation (J-Y Ji and E D Held, 2008 Phys. Plasmas 15 102 101). For verification of the formulas, example calculations for several lowest order moments are presented. The collisional moments can be applied in the derivations of Braginskii and integral (nonlocal) closures for arbitrary relative flow velocity between electrons and ions.

Monomial generators of complete planar ideals

We provide an algorithm that computes a set of generators for any complete ideal in a smooth complex surface. More interestingly, these generators admit a presentation as monomials in a set of maximal contact elements associated to the minimal log-resolution of the ideal. Furthermore, the monomial expression given by our method is an equisingularity invariant of the ideal. As an outcome, we provide a geometric method to compute the integral closure of a planar ideal and we apply our algorithm to some families of complete ideals.

On asymptotic depth of integral closure filtration and an application

Let (A,m) be an analytically unramified formally equidimensional Noetherian local ring with depthA≥2. Let I be an m-primary ideal and set I⁎ to be the integral closure of I. Set G⁎(I)=⨁n≥0(In)⁎/(In+1)⁎ be the associated graded ring of the integral closure filtration of I. We prove that depthG⁎(In)≥2 for all n≫0. As an application we prove that if A is also an excellent normal domain containing an algebraically closed field isomorphic to A/m then there exists s0 such that for all s≥s0 and J is an integrally closed ideal strictly containing (ms)⁎ then we have a strict inequality μ(J)<μ((ms)⁎) (here μ(J) is the number of minimal generators of J).

Descent of vector bundles under wildly ramified extensions

Given an irreducible normal Noetherian scheme and a finite Galois extension of the field of rational functions, we discuss the comparison of the categories of vector bundles on the scheme and equivariant vector bundles on the integral closure in the extension. This is well understood in the tame case (geometric stabilizer groups of order invertible in the local rings), so we focus on the wild (non-tame) case, which may be reduced to the case of cyclic extensions of prime order. In this case, under an additional flatness hypothesis, we give a characterization of the equivariant vector bundles that arise by base change from vector bundles on the scheme.

Real and complex integral closure, Lipschitz equisingularity and applications on square matrices

Real and complex integral closure, Lipschitz equisingularity and applications on square matrices

Numerical fracture analysis for chemo-mechanical coupling problems in multilayered porous media

The chemo-mechanical coupling fracture analysis is carried out numerically in this paper for the multilayered porous media. A chemo-mechanical extended layerwise method (CM-XLWM) is developed for the multilayered porous media with multiple interface and transverse cracks. The multiple interface cracks are simulated by the strong and weak discontinuous functions in the displacement assumptions along the thickness direction, whilst the transverse cracks are modeled by the extended finite element method (XFEM) in the in-plane displacement discretization. The mixed chemo-mechanical variational principle is employed to derive the Euler equations and the discrete formulations. Since the displacement and chemical fields are solved simultaneously, a fully coupled time integration method is developed. The interaction integral method and Virtual Crack Closure Technique are used to calculate the stress intensity factors and energy release rates, respectively. Several numerical examples are studied to demonstrate the capabilities of the CM-XLWM in static response analyses, stress intensity factor calculations and energy release rate for the multilayered porous media.

Some results about complete and analytic local rings

In this paper, we prove some results about inclusions of complete (or analytic) local rings. The main result says that if A⊆B are equicharacteristic zero complete (or analytic) local domains with the same uncountable residue field, such that the map Spec B→Spec A is surjective, then the integral closure of A in B is a finite A-module. In particular, if A⊆B is a pure extension of complete (or analytic) local domains, then the integral closure of A in B is a finite A-module.

SCC-Cryptosystem on an algebraic closure ring

The aim of this paper is to bring together two areas in which we propose a cryptosystem on a commutative ring based on the concept of integral basis in number theory. This idea goes back at least as far as [6] and [2]. We are interested in finding a commutative and associative ring denoted by ℤ q [α, β] from the integral closure QL of sextic extension L = ℚ(α, β), on which we constructed a new SCC-Cryptosystem.

On the reduction numbers of monomial ideals

Let [Formula: see text] be a monomial ideal in a polynomial ring with two indeterminates over a field. Assume [Formula: see text] is contained in the integral closure of some ideal that is generated by two elements from the generating set of [Formula: see text]. We produce sharp upper bounds for each of the reduction number and the Ratliff–Rush reduction number of the ideal [Formula: see text]. Under certain hypotheses, we give the exact values of these reduction numbers, and we provide an explicit method for obtaining these sharp upper bounds.