Martin Boundary of a Fractal Domain

A uniformly John domain is a domain intermediate between a John domain and a uniform domain. We determine the Martin boundary of a uniformly John domain D as an application of a boundary Harnack principle. We show that a certain self-similar fractal has its complement as a uniformly John domain. In particular, the complement of the 3-dimensional Sierpiski gasket is a uniform domain and its Martin boundary is homeomorphic to the Sierpiski gasket itself.     

Monoid Domain Constructions of Antimatter Domains

An integral domain without irreducible elements is called an antimatter domain. We give some monoid domain constructions of antimatter domains. Among other things, we show that if D is a GCD domain with quotient field K that is algebraically closed, real closed, or perfect of characteristic p > 0, then the monoid domain D[X; ?⁺] is an antimatter GCD domain. We also show that a GCD domain D is antimatter if and only if P^?1 = D for each maximal t-ideal P of D.     

Integral Closure of Graded Integral Domains

Let Γ be a torsion-free cancellative commutative monoid and let R = ?_α∈ΓR_α be a commutative Γ-graded ring. We show that if R is a graded Noetherian domain, then its integral closure is a graded Krull domain. This is a graded analog of the Mori–Nagata theorem. We also show that for a graded Strong Mori domain, its complete integral closure is a graded Krull domain but its integral closure is not necessarily a graded Krull domain.     

Gorenstein Prüfer整环的一些新刻画

设R是整环.众所周知,R是Prüfer整环当且仅当每个可除模是FP-内射模当且仅当每个h-可除模是FP-内射模.本文引进了一种新的Gorenstein FP-内射模,并且证明了R是Gorenstein Prüfer整环当且仅当每个可除模是Gorenstein FP-内射模,当且仅当每个h-可除模是Gorenstein FP-内射模.     

Straight Rings

A (commutative integral) domain is called a straight domain if A ? B is a prime morphism for each overring B of A; a (commutative unital) ring A is called a straight ring if A/P is a straight domain for all P ∈ Spec(A). A domain is a straight ring if and only if it is a straight domain. The class of straight rings sits properly between the class of locally divided rings and the class of going-down rings. An example is given of a two-dimensional going-down domain that is not a straight domain. The classes of straight rings, of locally divided rings, and of going-down rings coincide within the universe of seminormal weak Baer rings (for instance, seminormal domains). The class of straight rings is stable under formation of homomorphic images, rings of fractions, and direct limits. The “straight domain" property passes between domains having the same prime spectrum. Straight domains are characterized within the universe of conducive domains. If A is a domain with a nonzero ideal I and quotient field K, characterizations are given for A ? (I: _K I) to be a prime morphism. If A is a domain and P ∈ Spec(A) such that A _P is a valuation domain, then the CPI-extension C(P) := A + PA _P is a straight domain if and only if A/P is a straight domain. If A is a going-down domain and P ∈ Spec(A), characterizations are given for A ? C(P) to be a prime morphism. Consequences include divided domain-like behavior of arbitrary straight domains.     

Factorization in Integral Domains,IV

Let D be an integral domain such that every nonzero nonunit of D is a finite product of irreducible elements. In this article, we introduce and study several unifying concepts for the theory of nonunique factorization in D. They give a new way to measure, in some sense, how far an half-factorial domain (resp., bounded factorization domain, atomic domain) D is from being a UFD (resp., finite factorization domain, Cohen–Kaplansky domain) based on equivalence relations on the set of irreducible elements of D.     

Singularities of the structure of two-sided ideals of a domain of elementary divisors

We prove that, in a domain of elementary divisors, the intersection of all nontrivial two-sided ideals is equal to zero. We also show that a Bézout domain with finitely many two-sided ideals is a domain of elementary divisors if and only if it is a 2-simple Bézout domain.     

On power series rings over valuation domains

Let V be a valuation domain and V[[X]] be the power series ring over V. In this paper, we show that if V[[X]] is a locally finite intersection of valuation domains, then V is an SFT domain and hence a discrete valuation domain. As a consequence, it is shown that the power series ring V[[X]] is a Krull domain if and only if V[[X]] is a generalized Krull domain if and only if V[[X]] is an integral domain of Krull type (or equivalently, a PvMD of finite t-character) if and only if V is a discrete valuation domain with Krull dimension at most one.     

Prüfer's ascent result via Inc

Let R be an integral domain whose integral closure is a Pr¨fer domain. It is proved that R ? T has the incomparability property for each integral domain T which contains R and is algebraic over R. As a corollary, one has a new proof of Pr¨fer's ascent result, which states that if R is as above and T is the integral closure of R in some field containing R, then T is a Pr¨fer domain.     

An improvement on geometrical parameterizations by transfinite maps

We present a method to generate a non-affine transfinite map from a given reference domain to a family of deformed domains. The map is a generalization of the Gordon–Hall transfinite interpolation approach. It is defined globally over the reference domain. Once we have computed some functions over the reference domain, the map can be generated by knowing the parametric expressions of the boundaries of the deformed domain. Being able to define a suitable map from a reference domain to a desired deformation is useful for the management of parameterized geometries.     

Complexity spaces as quantitative domains of computation

We study domain theoretic properties of complexity spaces. Although the so-called complexity space is not a domain for the usual pointwise order, we show that, however, each pointed complexity space is an ω-continuous domain for which the complexity quasi-metric induces the Scott topology, and the supremum metric induces the Lawson topology. Hence, each pointed complexity space is both a quantifiable domain in the sense of M. Schellekens and a quantitative domain in the sense of P. Waszkiewicz, via the partial metric induced by the complexity quasi-metric.     

Parallel domain decomposition procedures of improved D-D type for parabolic problems

Two parallel domain decomposition procedures for solving initial-boundary value problems of parabolic partial differential equations are proposed. One is the extended D-D type algorithm, which extends the explicit/implicit conservative Galerkin domain decomposition procedures, given in [5], from a rectangle domain and its decomposition that consisted of a stripe of sub-rectangles into a general domain and its general decomposition with a net-like structure. An almost optimal error estimate, without the factor H^−1/2 given in Dawson-Dupont’s error estimate, is proved. Another is the parallel domain decomposition algorithm of improved D-D type, in which an additional term is introduced to produce an approximation of an optimal error accuracy in L²-norm.     

拟共形映射和弱John域

作为John域的推广,本文定义了弱John域,并讨论了弱John域与拟圆、弱John域与拟共形 映射之间的关系,得到(1)若(?)。中的Jordan域D和它的外部 均是弱John域,则D 是拟圆;(2)R2中的弱John域是拟共不变的;(3)R2中的有界拟圆必是弱John域．最后构造例子 说明R2中的无界拟圆不一定是弱John域．     

Centralizers in Domains of Gelfand-Kirillov Dimension 2

Given an affine domain of Gelfand–Kirillov dimension 2over an algebraically closed field, it is shown that the centralizerof any non-scalar element of this domain is a commutative domainof Gelfand–Kirillov dimension 1 whenever the domain isnot polynomial identity. It is shown that the maximal subfieldsof the quotient division ring of a finitely graded Goldie algebraof Gelfand–Kirillov dimension 2 over a field F all havetranscendence degree 1 over F. Finally, centralizers of elementsin a finitely graded Goldie domain of Gelfand–Kirillovdimension 2 over an algebraically closed field are considered.In this case, it is shown that the centralizer of a non-scalarelement is an affine commutative domain of Gelfand–Kirillovdimension 1. 2000 Mathematics Subject Classification 16P90.     

Decomposition and removability properties of John domains

In this paper we characterize John domains in terms of John domain decomposition property. In addition, we also show that a domain D in ℔ ⁿ is a John domain if and only if D\P is a John domain, where P is a subset of D containing finitely many points of D. The best possibility and an application of the second result are also discussed.     

Pseudoconvexity and Gromov hyperbolicity

We give an estimate for the distance functions related to the Bergman, Carathéodory, and Kobayashi metrics on a bounded strictly pseudoconvex domain with C²-smooth boundary. Our formula relates the distance function on the domain with the Carnot- Carathéodory metric on the boundary. As a corollary we conclude that the domain equipped with the any of the standard invariant distances is hyperbolic in the sense of Gromov. When the boundary of the domain is C³-smooth, our estimate is exact up to a fixed additive term.     

Block-diagonal reduction of matrices over an <Emphasis Type="Italic">n</Emphasis>-simple Bézout domain (n ≥ 3)

It is known that a simple Bézout domain is the domain of elementary divisors if and only if it is 2-simple. The block-diagonal reduction of matrices over an n -simple Bézout domain (n ≥ 3) is realized.     

Domains whose overrings satisfy accp

An integral domain D satisfies ACC on principal ideals (ACJCP) if there does not exist an infinite strictly ascending chain of principal ideals of D. Any Noetherian domain, in particular any Dedekind domain, satisfies ACCP. In this note we prove the following theorem: Let D be an integral domain. Then the integral closure of D is a Dedekind domain if and only if every overring of D (ring between D and its quotient field) satisfies ACCP.     

Finite character of finitely stable domains

A commutative domain is finitely stable if every nonzero finitely generated ideal is stable, i.e. invertible over its endomorphism ring. A domain satisfies the local stability property provided that every locally stable ideal is stable.We prove that a finitely stable domain satisfies the local stability property if and only if it has finite character, that is every nonzero ideal is contained in at most finitely many maximal ideals. This result allows us to answer the open problem of whether every Clifford regular domain is of finite character.     

Products of idempotent matrices over integral domains

We say that a ring R has the idempotent matrices property if every square singular matrix over R is a product of idempotent matrices. It is known that every field, and more generally, every Euclidean domain has the idempotent matrices property. In this paper we show that not every integral domain has the idempotent matrices property and that if a projective free ring has the idempotent matrices property then it must be a Bezout domain. We also show that a principal ideal domain has the idempotent matrices property if and only if every fraction a/b with b≠0 has a finite continued fraction expansion. New proofs are also provided for the results that every field and every Euclidean domain have the idempotent matrices property.