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     

R+ˆ is surprisingly an integral domain

It is shown that if $(R,\mathfrak{m},k)$ is a complete local domain with $\mathrm{char}k=p>0$ and $R^{+}$ is its integral closure in an algebraic closure of the quotient field, then both the $\mathfrak{m}$-adic and p-adic completions of $R^{+}$ are integral domains. More generally, this theorem remains true if the completeness assumption is relaxed to allow R to be an analytically irreducible Henselian local ring. It is also shown that these rings, which are Cohen-Macaulay R-modules (even balanced in the $\mathfrak{m}$-adic case), will have dimension larger than the dimension of R unless $\dim ?R\le 1$.     

On the set of <Emphasis Type="Italic">t</Emphasis>-linked overrings of an integral domain

Let R be an integral domain with quotient field L. An overring T of R is t-linked over R if I ⁻¹ = R implies that (T : IT)  =  T for each finitely generated ideal I of R. Let O _t (R) denotes the set of all t-linked overrings of R and O(R) the set of all overrings of R. The purpose of this paper is to study some finiteness conditions on the set O _t (R). Particularly, we prove that if O _t (R) is finite, then so is O(R) and O _t (R) = O(R), and if each chain of t-linked overrings of R is finite, then each chain of overrings of R is finite. This yields that the t-linked approach is more efficient than the Gilmer’s treatment (Proc Am Math Soc 131:2337–2346, 2002). We also examine the finiteness conditions in some Noetherian-like settings such as Mori domain, quasicoherent Mori domain, Krull domain, etc. We establish a connection between O _t (R) and the set of all strongly divisorial ideals of R and we conclude by a characterization of domains R that are t-linked under all their overrings. This work was funded by KFUPM under Project # FT/18-2005.     

TRANSCENDENTAL AND ALGEBRAIC EXTENSIONS OF COMMUTATIVE RINGS

Abstract

Transcendental and algebraic elements over commutative rings are defined. Rings with zero nil radical are considered. For a transcendental over R, necessary and sufficient conditions are derived for elements of R[α] to be algebraic or transcendental over R. For R a ring with identity and a finite number of minimal prime ideals, necessary and sufficient conditions are given for any element in a unitary overring of R to be algebraic or transcendental over R. It is proved that if α is algebraic Over R, so is every element of R[α]. It is show that if R is Noetherian, β is algebraic over R[α] and α is algebraic over R, then, under certain conditions, β is algebraic over R. If R has a finite number of minimal prime ideals, P₁,…,P_k, which are pairwise comaximal, then if t is transcendental over R, R[t] can be obtained by adjoining k algebraic elements a_i over R to R whose defining polynomials are in P_i [x], and conversely, if such elements are adjoined to R, they generate an element transcendental over R.     

Example of an interpolation domain

Cahen et al. (J. Algebra 225 (2000) 794), have defined a domain to be an interpolation domain if, essentially, Lagrange interpolation can be done using integer-valued polynomials. They prove results indicating that in some cases every overring of an interpolation domain is again an interpolation domain, and they ask whether the statement holds in general. In the present note, we provide a counterexample to the general statement.     

On the singular Bochner-Martinelli integral

The Bochner-Martinelli (B.-M.) kernel inherits, forn2, only some of properties of the Cauchy kernel in . For instance it is known that the singular B.-M. operatorM _n is not an involution forn2. M. Shapiro and N. Vasilevski found a formula forM ₂ ² using methods of quaternionic analysis which are essentially complex-twodimensional. The aim of this article is to present a formula forM _n ² for anyn2. We use now Clifford Analysis but forn=2 our formula coincides, of course, with the above-mentioned one.     

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     

A note on coring extensions

A notion of a coring extension is defined and it is shown to be equivalent to the existence of an additive functor between comodule categories that factorises through forgetful functors. This correspondence between coring extensions and factorisable functors is illustrated by functors between categories of descent data. A category in which objects are corings and morphisms are coring extensions is also introduced.

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Sunto Si fornisce una definizione per estensioni di coanelli e si dimostra l'equivalenza di tale definizione con l'esistenza di un funtore additivo tra categorie di comoduli che si fattorizzi attraverso il funtore dimenticante. Questa corrispondenza tra estensioni di coanelli e funtori fattorizzabili è illustrata da funtori tra categorie di discesa. Si introduce inoltre una categoria i cui oggetti sono coanelli e i morfismi sono estensioni di coanelli.

     

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     

Maximal Betti numbers of Cohen–Macaulay complexes with a given <Emphasis Type="Italic">f</Emphasis>-vector

Given the f-vector f = (f₀, f₁, . . .) of a Cohen–Macaulay simplicial complex, it will be proved that there exists a shellable simplicial complex Δ_f with f(Δ_f) = f such that, for any Cohen–Macaulay simplicial complex Δ with f(Δ) = f, one has for all i and j, where f(Δ) is the f-vector of Δ and where β _ij (I _Δ) are graded Betti numbers of the Stanley–Reisner ideal I _Δ of Δ. The first author is supported by JSPS Research Fellowships for Young Scientists. Received: 23 January 2006     

Structure of commutative cancellative integral residuated lattices on (0, 1]

ΠMTL-algebras were introduced as an algebraic counterpart of the cancellative extension of monoidal t-norm based logic. It was shown that they form a variety generated by ΠMTL-chains on the real interval [0, 1]. In this paper the structure of these generators is investigated. The results illuminate the structure of cancellative integral commutative residuated chains, because every such algebra belongs to the quasivariety generated by the zero-free subreducts on (0, 1] of all ΠMTL-chains on [0, 1]. The work of the author was partly supported by the grant No. A100300503 of the Grant Agency of the Academy of Sciences of the Czech Republic and partly by the Institutional Research Plan AV0Z10300504.     

On criteria by Dedekind and Ore for integral ring extensions

Dedekinds criterion gives a necessary and sufficient condition for the integral closure of a Dedekind ring in a finite (separable) extension to be monogenic (generated by one element). We shall demonstrate its usefulness in applications, and we shall prove a slightly stronger result which avoids any separability assumption. We similarly interprete further classical work by Ore.Received: 19 July 2004     

The minimum distance of sets of points and the minimum socle degree

Let K be a field of characteristic 0. Let be a reduced finite set of points, not all contained in a hyperplane. Let be the maximum number of points of Γ contained in any hyperplane, and let . If I⊂R=K[x₀,…,x_n] is the ideal of Γ, then in Tohaˇneanu (2009) [12] it is shown that for n=2,3, d(Γ) has a lower bound expressed in terms of some shift in the graded minimal free resolution of R/I. In these notes we show that this behavior holds true in general, for any n≥2: d(Γ)≥A_n, where A_n=min{a_i−n} and ⊕_iR(−a_i) is the last module in the graded minimal free resolution of R/I. In the end we also prove that this bound is sharp for a whole class of examples due to Juan Migliore (2010) [10].     

Taylor complexes for Koszul boundaries

For all boundary modules of the Koszul complex of a monomial sequence we construct complexes, which we call Taylor complexes. For a monomial d-sequences these complexes provide free resolutions of the boundary modules. Let M be the ideal generated by a monomial d-sequence. We use the Taylor complexes to construct minimal free resolutions of the Rees algebra and the associated graded ring of M. Received: 13 November 1997 / Revised version: 6 March 1998     

Normal integral bases in quadratic and cyclic cubic extensions of quadratic fields

Let K be a number field and let G be a finite abelian group. We call K a Hilbert-Speiser field of type G if, and only if, every tamely ramified normal extension L/K with Galois group isomorphic to G has a normal integral basis. Now let C₂ and C₃ denote the cyclic groups of order 2 and 3, respectively. Firstly, we show that among all imaginary quadratic fields, there are exactly three Hilbert-Speiser fields of type $C_{2}: \mathbb{Q}(\sqrt {m})$, where $m \in \{-1, -3, -7\}$. Secondly, we give some necessary and sufficient conditions for a real quadratic field $K = \mathbb{Q}(\sqrt {m})$ to be a Hilbert-Speiser field of type C₂. These conditions are in terms of the congruence class of m modulo 4 or 8, the fundamental unit of K, and the class number of K. Finally, we show that among all quadratic number fields, there are exactly eight Hilbert-Speiser fields of type $C_{3}: \mathbb{Q}(\sqrt {m})$, where $m \in \{-11,-3, -2, 2, 5, 17, 41, 89\}$.Received: 2 April 2002     

Self-tests for freeness over commutative artinian rings

We prove that the Auslander-Reiten conjecture holds for commutative standard graded artinian algebras, in two situations: the first is under the assumption that the modules considered are graded and generated in a single degree. The second is under the assumption that the algebra is generic Gorenstein of socle degree 3.     

Power series extensions of half-factorial domains

An integral domain is said to be a half-factorial domain (HFD) if every non-zero element a that is not a unit may be factored into a finite product of irreducible elements, while any other such factorization of a has the same number of irreducible factors. While it is known that a power series extension of a factorial domain need not be factorial, the corresponding question for HFD has been open. In this paper we show that the answer is also negative. In the process we answer in the negative, for HFD, an open question of Samuel for factorial domains by showing that for certain quadratic domains R, and independent variables, Y and T, R[[Y]][[T]] is not HFD even when R[[Y]] is HFD. The proof hinges on Samuel’s theorem to the effect that a power series, in finitely many variables, over a regular factorial domain is factorial.     

A characterization of Cohen-Macaulay modules and local cohomology

Let R be a commutative Noetherian ring, and let N be a non-zero finitely generated locally quasi-unmixed R-module. In this paper, the main result asserts that N is Cohen-Macaulay if and only if, for any N-proper ideal I of R generated by height_N I elements, the set of asymptotic primes of I with respect to N is equal to the set of presistent primes of I with respect to N. In addition, some applications about local cohomology are included. Received: 3 July 2005