Showing papers on "Ring (mathematics) published in 1987"
01 Jan 1987
269 citations
TL;DR: In this paper, the authors view the cyclic homology of an algebra as a composite functor, and propose a formula for HC,(A @A' ) that involves the periodicity operator S and the Hochschild groups.
235 citations
TL;DR: In this paper, the Bethe-ansatz solution, the exact factorized complete S -matrix and the particle spectrum for a two-dimensional chiral field on principal manifolds associated with the classical Lie groups SU( k + 1), SO(2 k ), SO( 2 k+ 1), Sp(2k ) are presented.
181 citations
TL;DR: In this paper, it was shown that a semiprime ring R must have a nontrivial central ideal if it admits an appropriate endomorphism or derivation which is centralizing on some one-sided ideal.
Abstract: Let R be a ring with center Z, and S a nonempty subset of R. A mapping F from R to R is called centralizing on S if [x, F(x)] ∊ Z for all x ∊ S. We show that a semiprime ring R must have a nontrivial central ideal if it admits an appropriate endomorphism or derivation which is centralizing on some nontrivial one-sided ideal. Under similar hypotheses, we prove commutativity in prime rings.
164 citations
TL;DR: In this paper, Anderson-Fuller et al. extended the theory of Morita equivalence to rings without identity, and showed that a set of commuting idempotents is equivalent to a ring with unitary left modules.
Abstract: In the paper [1] Abrams made a first step in extending the theory of Morita equivalence to rings without identity. He considered rings in which a set of commuting idempotents is given such that every element of the ring admits one of these idempotents as a two-sided unit, and the categories of all left modules over these rings which are unitary in a natural sense. He proved that two such module categories over the rings R and S, say, are equivalent if and only if there exists a unitary left i?-module P which is a generator, the direct limit of a given kind of system of finitelygenerated projective modules, and such that S is isomorphic to the ring of certain endomorphisms of P. The aim of the present paper is to extend this theory in two ways: to cover a wider range of rings, and to transfer more of the classical Morita theory. Firstly, one can weaken the condition of commutativity of the idempotents in question: it sufficesto require that any two of them have a common upper bound under the natural partial order (i.e., any two elements of the ring admit a common two-sided identity), a condition which is fulfilledby all regular rings (regular in the sense of Neumann). Whenever one has such a system of idempotents, then any larger system, in particular, the set of all idempotents, is also such, which is not the case for the systems of Abrams. Secondly, by a suitable modification of some homological lemmas we obtain also the two-sided characterizations of Morita equivalence, arriving thus at a complete analogy to the classicalcase of rings with identity. Our presentation is a combination of those in Anderson-Fuller [2], §§21-22,and Bass [5] (see also [6], Chapter II). This machinery allows us to avoid the elaborate construction of Abrams. As examples we describe, among others, those rings with local units which are Morita equivalent to division rings and primary rings, respectively. The Rees matrix rings studied in [4] turn out to have a natural place in this theory. The theory we present here is a counterpart of the theory of Morita duality developed by Yamagata [10]. On the one hand, we shall use the same modified Hom-functors but for projective and not injective modules, and on the
129 citations
TL;DR: In this article, a prime ring with the centerZ and the extended centroidC is considered, and a polynomial over C in noncommuting variablesx>>\s 1, …,..., X>>\s n ∈ the ring of all 2 × 2 matrices over GF(2) is considered.
Abstract: SupposeR is a prime ring with the centerZ and the extended centroidC. Letp(x
1, …,x
n) be a polynomial overC in noncommuting variablesx
1, …,x
n. LetI be a nonzero ideal ofR andA be the additive subgroup ofRC generated by {p(a
1, …,a
n):a
1, …,a
n ∈I}. Then eitherp(x
1, …,x
n) is central valued orA contains a noncentral Lie ideal ofR except in the only one case whereR is the ring of all 2 × 2 matrices over GF(2), the integers mod 2.
83 citations
Journal Article•
01 Nov 1987
TL;DR: In this article, Sammons describes the ruin waiting for almost all those ill-advised enough to become professional boxers and confirms all the legends, of crime, of swindling, of the miserable economic rewards allotted to the vast majority of fighters.
Abstract: "Carefully documents the ruin waiting for almost all those ill-advised enough to become professional boxers. He confirms all the legends, of crime, of swindling, of the miserable economic rewards allotted to the vast majority of fighters...the traditional racism of the American ring...No one, reading Sammons, can doubt that it is evil." - "Times Literary Supplement."
81 citations
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TL;DR: In this article, it was shown that any torsion-free p-reduced, p-cotorsion free, commutative ring S may be embed88 0021-8693/87 $3.00
71 citations
TL;DR: In this paper, error analysis methods for computer simulations are reviewed and applied to a Brownian dynamics simulation of a ring polymer, and error analysis is applied to the simulation of the ring polymer.
Abstract: Error analysis methods for computer simulations are reviewed and applied to a Brownian dynamics simulation of a ring polymer.
62 citations
TL;DR: In this article, the symmetric Martindale ring of quotients for various classes of prime rings is discussed and computed. In particular, free and group algebras are considered.
54 citations
TL;DR: In this paper, the authors give a criterion for a projective Cohen-Macaulay variety X to have rational singularity when X does not have a singularity, and they give an application to multi-cones based on Schubert varieties.
Abstract: Let ~q~ 1 . . . . , 2 ' , be invertible sheaves on a variety X. We may form the multigraded ring A = ~F(X, | where the vector (m) ranges in N". Then C = Spec(A) is a multi-cone. If X is a projective Cohen-Macaulay variety, n = l and L~'l is very ample, Serre has given a criterion for C to be Cohen-Macaulay in terms of the cohomology of sheaves on X. We know no reasonable analog of this result when n > 1. On the other hand we will give such a criterion for C to have rational singularity when X does. This is a stronger restriction on the singularities of C. Then we will give an application to multi-cones based on Schubert varieties. Let G be a reductive algebraic group, B a Borel subgroup and X ~G/B a Schubert variety. Let s176 &~ ..., 5q~ be the line bundles on G/B corresponding to the fundamental weights. An arbitrary line bundle corresponding to a dominant weight has the form
08 Jun 1987
TL;DR: A model from which a superlinear speedup can be deduced is described, based on the fact that in the average the solutions are distributed nonuniformly in the case of the satisfiability problem.
Abstract: We have implemented a backtracking strategy for the satisfiability problem on a ring of processors and we observed a superlinear speedup in the average. In this paper we describe a model from which this superlinear speedup can be deduced. The model is based on the fact that in the average the solutions are distributed nonuniformly in the case of the satisfiability problem. To our knowledge this phenomenon was not used before in the analysis of algorithms.
TL;DR: In this paper, the authors show that the T-equivariant K-theory of the Borel subgroup of a Kac-Moody group admits operators similar to the Demazure operators defined on A(T).
Abstract: Let G be a Kac—Moody group with Borel subgroup B and compact maximal torus T. Analogous to Kostant and Kumar [Kostant, B. & Kumar, S. (1986) Proc. Natl. Acad. Sci. USA 83, 1543-1545], we define a certain ring Y, purely in terms of the Weyl group W (associated to G) and its action on T. By dualizing Y we get another ring Ψ, which, we prove, is “canonically” isomorphic with the T-equivariant K-theory KT(G/B) of G/B. Now KT(G/B), apart from being an algebra over KT(pt.) ≈ A(T), also has a Weyl group action and, moreover, KT(G/B) admits certain operators {Dw}w[unk]W similar to the Demazure operators defined on A(T). We prove that these structures on KT(G/B) come naturally from the ring Y. By “evaluating” the A(T)-module Ψ at 1, we recover K(G/B) together with the above-mentioned structures. We believe that many of the results of this paper are new in the finite case (i.e., G is a finite-dimensional semisimple group over C) as well.
TL;DR: In this paper, the use of "dual stream functions" in selecting a single approximating velocity field from a "sufficiently wide" orthogonalizing family is demonstrated.
TL;DR: In this paper, the pole assignability problem over commutative rings is studied in the context of algebraic systems theory. But the paper itself should be classified as pure commUTative algebra and we shall present it as such in the body of the text.
TL;DR: In this article, the authors studied the second layer condition of skew groups and group-graded rings and gave necessary and sufficient conditions for the skew group ring of a torsion-free nilpotent group to be simple.
TL;DR: The author examines two broadly supported ring designs, IEEE 802.5 and the Fiber Distributed Data Interface (FDDI), X3T9.5 currently emerging as local area network standards, and explores the basis of this interest in ring topologies.
Abstract: The author examines two broadly supported ring designs, IEEE 802.5 and the Fiber Distributed Data Interface (FDDI), X3T9.5 currently emerging as local area network (LAN) standards, and explores the basis of this interest in ring topologies. The advantages that ring designs offer are presented, and on this basis a proliferation of rings is envisioned for the next decade.
TL;DR: The use of vectors simplifies the representation of the molecular structures resulting from the ring puckering of asymmetric five-membered rings and has been used to calculate the one-dimensional ring-puckering potential functions for these molecules in dimensional form.
TL;DR: The quadratic residue number system (QRNS) as discussed by the authors is a generalization of the classical residue number number system, which is used for complex convolution over finite rings.
Abstract: Very recently, the quadratic residue number system (QRNS) has been introduced [8], [9]. It is, in fact, a rediscovery of earlier work [29]. The QRNS is obtained from a mapping of Gaussian integers over a finite ring to a ring of conjugate elements. This conjugate ring has the remarkable property that both addition and multiplication are performed componentwise. The operations are performed over subrings, isomorphic to the conjugate ring, and the results are mapped to the conjugate ring via the Chinese remainder theorem (CRT). The QRNS has since been generalized for any type of moduli set, with an inherent dynamic range reduction, and has been termed the quadratic-like residue number system (QLRNS) [7]. An alternate form removes the dynamic range reduction but requires an increase in multiplications from two to three; this system has been termed the modified quadratic residue number system (MQRNS) [3], [4]. This tutorial paper is a companion to another paper in this special issue [32]; it consolidates work on quadratic system implementations, with special emphasis on the modified system. The paper discusses, in some detail, the quadratic implementation of the two forms of complex convolution, which are naturally defined over finite rings, or fields, and which incur no scaling overhead. A new notation is introduced that eliminates confusion over the several mappings that are required in the quadratic representation, and initial work on the implementation of finite ring computational elements, suitable for VLSI fabrication, is also presented.
TL;DR: Models for local area networks of the slotted ring style of architecture of architecture are developed and evaluated and a limited comparison between the Cambridge Ring and another ring architecture—the token ring—is carried out.
Abstract: Models for local area networks of the slotted ring style of architecture are developed and evaluated. The hardware protocol is modeled using a BCMP network. The Basic Block protocol of the Cambridge ring is modeled using an approximate solution method of the fixed-point type. A limited comparison between the Cambridge Ring and another ring architecture—the token ring—is carried out.
TL;DR: This paper generalizes the notion of a reduction ring by giving weaker axioms that characterize a wider class of rings and proves that the ring of integers modulo z, z an arbitrary not necessarily prime integer, is a reductionRing in this generalized sense.
TL;DR: In this article, the incomparability result of D. Passman and M. Bergman for the infinite cyclic group is generalized to the case of twisted differential operator rings of solvable Lie algebras.
TL;DR: The structure of the pure injective modules appearing in such a resolution is known and can be used to give information about minimal injective resolutions of flat modules, which can in turn give new proofs of results about the projective dimension of projective modules and at the same time sharpen these results as mentioned in this paper.
TL;DR: In this paper, it was shown that the weak global dimension of the ring of k-linear differential operators on a nonsingular affine algebraic variety over k is equal to the dimension of this variety.
Book•
01 Sep 1987
TL;DR: In this article, Atkin and Swinnerton-Dyer showed congruences for a large class of branched double coverings of arbitrary dimension and genus, defined over any ring which is flat and of finite type over Z.
Abstract: Introduction. In this note we show congruences, similar to those of Atkin and Swinnerton-Dyer [2, 6], for a large class of schemes, including branched double coverings of pN of arbitrary dimension and genus, defined over any ring which is flat and of finite type over Z. The results of sections 1-4 together yield the following theorem. THEOREM 0.1. Let K be a ring which is flat and offinite type over Z. Let R E K[T0, . . *, TN] be a homogeneous polynomial of degree 2d. Assume 2d > 2N > 0. Let 9C be the double covering of PZ given by the equation U2 = R (where U is a new variable of weight d). Let (P be a maximal ideal of K with residue field K/1@ of characteristic p and of order q = p f. Let e be an integer such that I < e ? p - 1 and p e (We.
01 Dec 1987
TL;DR: In this article, a totally definite quaternion algebra over a real algebraic number field and a ring of algebraic integers of a real number field is considered, and formulas for the massm and type numbert(G) of G and for the trace of the Eichler-Brandt matrixB(G, J) ofG and any integral idealJ ofM in terms of genus invariants ofG.
Abstract: LetA be a totally definite quaternion algebra over a totally real algebraic number fieldF andM be the ring of algebraic integers ofF. For anyM-orderG ofA we derive formulas for the massm(G) and the type numbert(G) of G and for the trace of the Eichler-Brandt matrixB(G, J) ofG and any integral idealJ ofM in terms of genus invariants ofG and of invariants ofF andJ. Applications to class numbers of quaternion orders and of ternary quadratic forms are indicated.