Showing papers on "Ring (mathematics) published in 1989"

Uniform self-stabilizing rings

Abstract: A self-stabilizing system has the property that, no matter how it is perturbed, it eventually returns to a legitimate configuration. Dijkstra originally introduced the self-stabilization problem and gave several solutions for a ring of processors in his 1974 Communications of the ACM paper. His solutions use a distinguished processor in the ring, which effectively acts as a controlling element to drive the system toward stability. Dijkstra has observed that a distinguished processor is essential if the number of processors in the ring is composite. We show, by presenting a protocol and proving its correctness, that there is a self-stabilizing system with no distinguished processor if the size of the ring is prime. The basic protocol uses T (n2) states in each processor when n is the size of the ring. We modify the basic protocol to obtain one that uses T (n2/ln n) states.

Review of ring perception algorithms for chemical graphs

Abstract: Current ring perception algorithms for use on chemical graphs concentrate on processing specific structures. In this review, the various published ring perception algorithms are classified according to the initial ring set obtained, and each algorithm or method of perception is described in detail. The final ring sets obtained are discussed in terms of their suitability for use in representing the ring systems in structurally explicit parts of generic chemical structures.

A Characterization of F -Regularity in Terms of F -Purity

Abstract: In recent years, some very interesting theorems have been proven independently using complex analytic techniques or, alternatively, using reduction to characteristic p techniques (relying on special properties of the Frobenius homomorphism). In particular, Hochster and Roberts [12] proved that the ring R G of invariants of a group G acting on a regular ring R is necessarily Cohen-Macaulay by an argument which exploits the fact that R G is a direct summand of R in characteristic 0 and that, therefore, after reduction to characteristic p, the Frobenius homomorphism is especially well-behaved for “almost all p”. Not long after, using the Grauert-Riemenschneider vanishing theorem, Boutot [1] proved an even stronger result— in the affine and analytic cases, a direct summand (in characteristic 0) of a ring with rational singularity necessarily has a rational singularity.

A new algebraic approach to microlocalization of filtered rings

Abstract: Using the construction of the Rees ring associated to a filtered ring we provide a description of the microlocalization of the filtered ring by using only purely algebraic techniques. The method yields an easy approach towards the study of exactness properties of the microlocalization functor. Every microlocalization at a regular multiplicative Ore set in the associated graded ring can be obtained as the completion of a localization at an Ore set of the filtered ring.

A Method for the Study of Artinian Modules, With an Application to Asymptotic Behavior

Abstract: If N is a Noetherian module over the commutative ring R (throughout the paper, R will denote a commutative ring with identity), then the study of N in many contexts can be reduced to the study of a finitely generated module over a commutative Noetherian ring, because N has a natural structure as a module over R/(0 : N) and the latter ring is Noetherian. For a long time it has been a source of irritation to me that I did not know of any method which would reduce the study of an Artinian module A over the commutative ring R to the study of an Artinian module over a commutative Noetherian ring. However, during the MSRI Microprogram on Commutative Algebra, my attention was drawn to a result of W. Heinzer and D. Lantz [2, Proposition 4.3]; this proposition proves that if A is a faithful Artinian module over a quasi-local ring (R, M) which is (Hausdorff) complete in the M-adic topology, then R is Noetherian. It turns out that a generalization of this result provides a missing link to complete a chain of reductions by which one can, for some purposes, reduce the study of an Artinian module over an arbitrary commutative ring R to the study of an Artinian module over a complete (Noetherian) local ring; in the latter situation we have Matlis’s duality available, and this means that the investigation can often be converted into a dual one about a finitely generated module over a complete (Noetherian) local ring.

Symmetric bi-derivations on prime and semi-prime rings

Abstract: LetR be a ring. A bi-additive symmetric mappingD(.,.): R × R → R is called a symmetric bi-derivation if, for any fixedy ∈ R, a mappingx ↦ D(x, y) is a derivation. The purpose of this paper is to prove some results concerning symmetric bi-derivations on prime and semi-prime rings. We prove that the existence of a nonzero symmetric bi-derivationD(.,.): R × R → R, whereR is a prime ring of characteristic not two, with the propertyD(x, x)x = xD(x, x), x ∈ R, forcesR to be commutative. A theorem in the spirit of a classical result first proved by E. Posner, which states that, ifR is a prime ring of characteristic not two andD 1,D 2 are nonzero derivations onR, then the mappingx ↦ D 1(D 2 (x)) cannot be a derivation, is also presented.

Rings with zero intersection property on annihilators : zip rings.

Abstract: Zelmanowitz [12] introduced the concept of ring, which we call right zip rings, with the defining properties below, which are equivalent: (ZIP 1) If the right anihilator X^ of a subset X of R is zero, then X1^ = 0 for a finite subset X1 I X. (ZIP 2) If L is a left ideal and if L^ = 0, then L1^ = 0 for a finitely generated left ideal L1 I L. In [12], Zelmanowitz noted that any ring R satisfying the d.c.c. on anihilator right ideals (= dcc ^) is a right zip ring, and hence, so is any subring of R. He also showed by example that there exist zip rings which do not have dcc ^. In paragraph 1 of this paper, we characterize a right zip by the property that every injective right module E is divisible by every left ideal L such that L^ = 0. Thus, E = EL. (It suffices for this to hold for the injective hull of R). In paragraph 2 we show that a left and right self-injective ring R is zip iff R is pseudo-Frobenius (= PF). We then apply this result to show that a semiprime commutative ring R is zip iff R is Goldie. In paragraph 3 we continue the study of commutative zip rings.

Efficient elections in chordal ring networks

Abstract: We study the message complexity of the problem of distributively electing a leader in chordal rings. Such networks consist of a basic ring with additional links, the extreme cases being the oriented ring and the complete graph with a full sense of direction. We present a general election algorithm for these networks, and prove its optimality. As a corollary, we show thatO(logn) chords at each processor suffice to obtain an algorithm that uses at mostO(n) messages; this improves and extends a previous work, where an algorithm, also usingO(n) messages, was suggested for the case where alln-1 chords exist (the oriented complete network).

Theoretical aspects of ring perception and development of the extended set of smallest rings concept

Abstract: This paper considers the problems associated with processing planar (two-dimensional) representations of three-dimensional structures. To overcome the ambiguity of such representations, a new ring set is defined in terms of simple faces and cut faces. The concept of a cut-vertex graph is introduced to explain the combinatorial relationship between the number of simple faces and the number of planar embedments

A full uniform Artin-Rees theorem.

Abstract: This theorem provides a positive answer to a (generalized version of a) question of Eisenbud and Höchster [3], which arose in connection with their proof of a generalization of Zariski's Main Lemma on holomorphic functions. Our proof here depends on a development of the techniques of [7], where a weaker form of the present theorem was given. For a uniform Artin-Rees theorem in the analytic case (subject to a compactness restriction), with applications in Analysis, see [2], [4] (and the 1984 preprint mentioned in [4]).

Kaplansky’s problem on valuation rings

Abstract: The following theorem is proved in ZFC: there exist valuation rings which are not surjective homomorphic images of valuation domains. The proof relies on the existence of nonstandard divisible uniserial modules in ZFC. Let R be a valuation ring (in another terminology: a chained ring), i.e. a commutative ring with 1 in which the ideals form a chain under inclusion. A valuation domain is a domain that is a valuation ring at the same time. Kaplansky raised the question as to whether or not every valuation ring R can be obtained as a (surjective) homomorphic image of a valuation domain S. Under additional conditions on R, the answer is affirmative; see Ohm and Vicknair [3] and the literature cited there. On the other hand, Fuchs and Salce [2, p. 151] have given an explicit example for a valuation ring which can not be obtained in the indicated way. Their proof was based on the existence of nonstandard divisible uniserial modules by using Jensen's Diamond Principle which holds in the constructible universe but not in ZFC alone (Zermelo-Fraenkel set theory with the Axiom of Choice). Franzen and Gobel [1] pointed out that the slightly weaker hypothesis 2t4' < 2 ' suffices. (The existence of nonstandard uniserials was first established by Shelah [5] by using forcing argument; later he replaced it by an absoluteness result of stationary logic.) Our aim here is to prove in ZFC the existence of nonstandard divisible uniserial modules and the existence of valuation rings which are not obtainable as Received by the editors May 14, 1987 and, in revised form, January 24, 1988. 1980 Mathematics Subject Classification (1985 Revision). Primary 1 3A1 8, 1 3L05.

Appell rings in general relativity

Abstract: The authors analyse exact static solutions of the vacuum Einstein equations obtained from the Appell ring solution to the Laplace equation. They first perform a Newtonian analysis and show that the Appell solution may be interpreted as the field of a shell of infinite negative total mass bordered by a ring of infinite positive total mass. This situation can be reproduced in general relativity, but a different interpretation, not requiring the shell as a source, is also possible pasting two spacetimes through the ring. The superposition of a particle with an Appell ring is also considered and in this case the shell interpretation is found to be inconsistent.

Programmable conferencing module for ring arrays and switchable ring array networks

Abstract: The present invention pertains to a conferencing network. The conferencing network is comprised of a ring array comprised of K conferencing modules, where K is greater than or equal to 3 and is an integer. The K conferencing modules are capable of being in desired states. The conferencing network also includes means for controlling the ring array such that the conferencing modules are placed in the desired states. Switching stages comprised of switches can also be implemented into the conferencing network.

The graded Witt ring and Galois cohomology. II

Abstract: A primary problem in the theory of quadratic forms over a field F of characteristic different from two is to prove that the rings Hq*F and GWF are isomorphic. Here HqF = H*(Gal(Fq/F),Z/2Z)), where Fq is the quadratic closure of F, and GWF is the graded Witt nrng associated to the fundamental ideal of even dimensional forms in the Witt ring WF of F. In this paper, we assume we are given a field extension K of F such that WK is 'close' to WF or Hq*K is 'close' to Hq F . A method is developed to obtain information about these graded rings over F and its 2-extensions from information about the corresponding graded ring of K. This relative theory extends and includes the previously developed absolute case where K = Fq. Applications are also given to show that Hq F and GWF are isomorphic for a collection of fields anrsing naturally from the theory of abstract Witt rings.

Auslander algebras as quasi-hereditary algebras

Abstract: The notion of a quasi-hereditary algebra has been introduced by E. Cline, B. Parshall and L. Scott [7,2,5] in order to describe the so-called highest weight categories arising in the representation theory of Lie algebras and algebraic groups. Quasi-hereditary algebras are defined by the existence of a suitable chain of ideals, and the finite dimensional hereditary algebras are typical examples. In [3], also finite dimensional algebras of global dimension 2 are shown to be quasi-hereditary. Thus, the Auslander algebras are quasi-hereditary. Recall that the Auslander algebras A can be constructed in the following way. Let R be a representation-finite finite dimensional algebra; then A is the endomorphism algebra End(Affl), where M is a finite dimensional /{-module such that every indecomposable /{-module is isomorphic to a direct summand of M. We are going to introduce the notion of a splitting filtration on the class of all indecomposable /{-modules and show that in this way we obtain a heredity chain of ideals of A (see the definition below). Usually, there exist many splitting filtrations for a given R. Examples of splitting filtrations can be obtained from the Rojter measure, used by A. V. Rojter in his proof of the first Brauer-Thrall conjecture [6], or from the preprojective and preinjective partitions, introduced by M. Auslander and S. Smalo in [1]. Instead of dealing with finite dimensional algebras, we shall consider, more generally, semiprimary rings. Recall that an associative ring A with 1 is called semiprimary provided that its Jacobson radical TV is nilpotent and A/N is semisimple artinian. We say that an ideal J of A is a heredity ideal of A if P = J, JNJ = 0 and /, considered as a right ideal, is a projective ^-module. Following [2], a semiprimary ring A is said to be quasi-hereditary provided that there exists a chain