Minimal Overrings of an Integrally Closed Domain

Abstract

The main purpose of this paper is to characterize minimal overrings of an integrally closed domain R. We show that there exists a strong relationship between minimal overrings and the notion of ideal transforms. In particular, we prove that if T(M) = S(M) for each maximal ideal M, then there is a bijective correspondence between the set of invertible maximal ideals of R and the set of minimal overrings of R. This study enables us to produce several interesting applications concerning semi-local, Dedekind, Prüferian and Krull domains. Moreover, we investigate the spectrum of a minimal overring in comparison with the spectrum of R, and we determine whether the polynomial ring R[X ₁, X ₂,…, X _n ] has a minimal overring.     

Towards a classification of stable semistar operations on a Prüfer domain

We study stable semistar operations defined over a Prüfer domain, showing that, if every ideal of a Prüfer domain R has only finitely many minimal primes, every such closure can be described through semistar operations defined on valuation overrings of R.     

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.     

On Finite Saturated Chains of Overrings

If a domain R, with quotient field K, has a finite saturated chain of overrings from R to K, then the integral closure of R is a Prüfer domain. An integrally closed domain R with quotient field K has a finite saturated chain of overrings from R to K with length m ≥ 1 iff R is a Prüfer domain and |Spec(R)| =m + 1. In particular, we prove that a domain R has a finite saturated chain of overrings from R to K with length dim(R) iff R is a valuation domain and that an integrally closed domain R has a finite saturated chain of overrings from R to K with length dim (R) +1 iff R is a Prüfer domain with exactly two maximal ideals such that at most one of them fails to contain every non-maximal prime. The relationship with maximal non-valuation subrings is also established.     

Numerical characterizations of some integral domains

Let R be an integral domain. If R satisfies some finiteness conditions, an algorithm that produces the exact list of all distinct overrings of R is established. Then numerical characterizations of several classes of integral domains are obtained.     

N-free orders and minimal interval extensions

We investigate problems related to the set of minimal interval extensions of N-free orders. This leads us to a correspondence between this set for an arbitrary order and a certain set of its maximal N-free reductions. We also get a 1-1-correspondence between the set of linear extensions of an arbitrary order and the set of minimal interval extensions of the linegraph of that order. This has an algorithmic consequence, namely the problem of counting minimal interval extensions of an N-free order is #P-complete. Finally a characterization of all N-free orders with isomorphic root graph is given in terms of their lattice of maximal antichains; the lattices are isomorphic iff the root graphs agree.This work was supported by the PROCOPE Program. The first author is supported by the DFG.     

On a Field-Theoretic Invariant for Extensions of Commutative Rings

Abstract

If R ? T is an extension of (commutative integral) domains, Λ(T/R) is defined as the supremum of the lengths of chains of intermediate fields in the extension k _R (Q ∩ R) ? k _T (Q), where Q runs over the prime ideals of T. The invariant Λ(T/R) is determined in case R and T are adjacent rings and in case Spec(R) = Spec(T) as sets. It is proved that if R is a domain with integral closure R′, then Λ(T/R) = 0 for all overrings T of R if and only if R′ is a Prüfer domain such that Λ(R′/R) = 0. If R ? T are domains such that the canonical map Spec(T) → Spec(R) is a homeomorphism (in the Zariski topology), then Λ(T/R) is bounded above by the supremum of the lengths of chains of rings intermediate between R and T. Examples are given to illustrate the sharpness of the results.     

The number of overrings of an integrally closed domain

We establish in this work a result that gives the number of overrings for any integrally closed domain that has only finitely many overrings; then we provide an algorithm to compute this number. We end this paper with an open problem for integral domains that are not necessarily integrally closed.     

Umt-domains and domains with prüfer integral closure

An integral domain R is said to be a UMT-domain if uppers to zero in R[X) are maximal t-ideals. We show that R is a UMT-domain if and only if its localizations at maximal tdeals have Prüfer integral closure. We also prove that the UMT-property is preserved upon passage to polynomial rings. Finally, we characterize the UMT-property in certian pullback constructions; as an application, we show that a domain has Prüfer integral closure if and only if all its overrings are UMT-domains.     

Enneper immersions

We describe a method to construct minimal surfaces inR ³ which has many computational simplicities and prove that any immersed minimal surface inR ³ may be constructed by using that method. We also show that under certain finite hypotheses on the orders of the coordinate functions, among all conformal minimal immersions of the plane with Gauss mapg=e ^az+b only the parametrizations of helicoids and planes are proper embeddings.     

Resolutions of 2 and 3 Dimensional Rings of Invariants for Cyclic Groups

Let G be the cyclic group of order n, and suppose F is a field containing a primitive nth root of unity. We consider the ring of invariants F[W] ^G of a three dimensional representation W of G where G ? SL(W). We describe minimal generators and relations for this ring and prove that the lead terms of the relations are quadratic. These minimal generators for the relations form a Gröbner basis with a surprisingly simple combinatorial structure. We describe the graded Betti numbers for a minimal free resolution of F[W] ^G . The case where W is any two dimensional representation of G is also handled.     

Lamron ℓ-groups

The article introduces a new class of lattice-ordered groups. An ?-group G is lamron if Min(G)^?1 is a Hausdorff topological space, where Min(G)^?1 is the space of all minimal prime subgroups of G endowed with the inverse topology. It will be evident that lamron ?-groups are related to ?-groups with stranded primes. In particular, it is shown that for a W-object (G,u), if every value of u contains a unique minimal prime subgroup, then G is a lamron ?-group; such a W-object will be said to have W-stranded primes. A diverse set of examples will be provided in order to distinguish between the notions of lamron, stranded primes, W-stranded primes, complemented, and weakly complemented ?-groups.     

Some notes on densely continuous forms

We consider a special space of set-valued functions (multifunctions), the space of densely continuous forms D(X, Y) between Hausdorff spaces X and Y, defined in [HAMMER, S. T.—McCOY, R. A.: Spaces of densely continuous forms, Set-Valued Anal. 5 (1997), 247–266] and investigated also in [HOLá, L’.: Spaces of densely continuous forms, USCO and minimal USCO maps, Set-Valued Anal. 11 (2003), 133–151]. We show some of its properties, completing the results from the papers [HOLY, D.—VADOVIČ, P.: Densely continuous forms, pointwise topology and cardinal functions, Czechoslovak Math. J. 58(133) (2008), 79–92] and [HOLY, D.—VADOVIČ, P.: Hausdorff graph topology, proximal graph topology and the uniform topology for densely continuous forms and minimal USCO maps, Acta Math. Hungar. 116 (2007), 133–144], in particular concerning the structure of the space of real-valued locally bounded densely continuous forms D _p ^*(X) equipped with the topology of pointwise convergence in the product space of all nonempty-compact-valued multifunctions. The paper also contains a comparison of cardinal functions on D _p ^*(X) and on real-valued continuous functions C _p (X) and a generalization of a sufficient condition for the countable cellularity of D _p ^*(X). This work was supported by Science and Technology Assistance Agency under the contract No. APVT-51-006904 and by the Eco-Net (EGIDE) programme of the Laboratoire de Mathématiques de l’Université de Saint-Etienne (LaMUSE), France.     

Strikte Konvexität für Variationsprobleme auf demn-dimensionalen Torus

We consider a variational problem with an integrandF:R ⁿ ×R×R ⁿ →R that isZ-periodic in the firstn+1 variables and satisfies certain growth-conditions. By a recent result of Moser, there exist for every α∈R ⁿ minimal solutionsu:R ⁿ →R minimising ƒF(x, u(x), u _x (x)) dx with respect to compactly supported variations ofu and such that sup |u(x)-αx|<∞. Given such a minimal solutionu we define the average action (whereB _r is ther-ball around 0∈R ⁿ ) and show thatM(α) is indeed independent of the minimal solutionu satisfying sup |u(x)-αx|<∞. We prove that this average actionM(α) is strictly convex in α.      

Half-space type theorems in warped product spaces with one-dimensional factor

This work states some half-space type theorems in a warped product space of the form I ×_ρ M, where is an open interval and M is either a compact n-manifold, or a complete simply connected surface with constant curvature c ≤ 0. Such theorems generalize the classical half-space theorem for minimal surfaces in R ³, obtained by Hoffmann and Meeks (Invent Math 101:373–377, 1990), and recent results for surfaces contained in a slab of R ×_ρ M, obtained by Dajczer and Alías (Comment Math Helvetici 81:653–663, 2006).      

Bernstein type theorems with flat normal bundle

We prove Bernstein type theorems for minimal n-submanifolds in ℝn+p with flat normal bundle. Those are natural generalizations of the corresponding results of Ecker-Huisken and Schoen-Simon-Yau for minimal hypersurfaces. The research of the first author was supported by a Heisenberg fellowship of the DFG The research of the third author was partially supported by project # 973 of MSTC and SFECC Mathematics Subject Classification Primary (2000) 53C42     

τ-Categories II: Nakayama Pairs and Rejective Subcategories

We study Nakayama pairs in τ-categories, which are pairs of objects connected by a certain diagram of Auslander–Reiten sequences. Using them, we naturally introduce orderlikeτ-categories. Then we study rejective subcategories of τ-categories. A typical example of orderlike τ-categories is given by the category of lattices over an order. In this case, its rejective subcategories correspond bijectively to its overrings. Presented by K. Roggenkamp Mathematics Subject Classifications (2000) Primary: 16G30; secondary: 16E65, 16G70, 18E05. Osamu Iyama: Current address: Department of Mathematics, University of Hyogo, Himeji 671-2201, Japan. e-mail: iyama@sci.u-hyogo.ac.jp