Showing papers on "Ring (mathematics) published in 1985"

Representations of Rings Over Skew Fields

Abstract: Part I Homomorphisms to simple artinian rings: 1 Hereditary rings and projective rank functions 2 The coproduct theorems 3 Projective rank functions on ring coproducts 4 Universal localisation 5 Universal homomorphisms from hereditary to simple artinian rings 6 Homomorphisms from hereditary to von Neumann regular rings 7 Homomorphisms from rings to simple artinian rings Part II Skew subfields of simple artinian coproducts: 8 The centre of the simple artinian coproduct 9 Finite dimensional divisions subalgebras of skew field coproducts 10 The universal bimodule of derivations 11 Commutative subfields and centralisers in skew held coproducts 12 Characterising universal localisations at a rank function 13 Bimodule amalgam rings and Artin's problem References Index

Computing on an anonymous ring

Abstract: The computational capabilities of a system of n indistinguishable (anonymous) processors arranged on a ring in the synchronous and asynchronous models of distributed computation are analyzed. A precise characterization of the functions that can be computed in this setting is given. It is shown that any of these functions can be computed in O(r?) messages in the asynchronous model. This is also proved to be a lower bound for such elementary functions as AND, SUM, and Orientation. In the synchronous model any computable function can be computed in O(n log n) messages. A ring can be oriented and start synchronized within the same bounds. The main contribution of this paper is a new technique for proving lower bounds in the synchronous model. With this technique tight lower bounds of O(nlogn) (for particular n) are proved for XOR, SUM, Orientation, and Start Synchronization. The technique is based on a string-producing mechanism from formal language theory, first introduced by Thue to study square-free words. Two methods for generalizing the synchronous lower bounds to arbitrary ring sizes are presented.

A Transfer-Function Approach to Linear Time-Varying Discrete-Time Systems

Abstract: In the first part of the paper a transfer-function approach is developed for the class of linear time-varying discrete-time systems. The theory is specified in terms of skew (noncommutative) rings of polynomials and formal power series, both with coefficients in a ring of time functions. The transfer-function matrix is defined to be a matrix whose entries belong to a skew ring of formal power series. It is shown that various system properties, such as asymptotic stability, can be characterized in terms of the skew-ring framework. In the last part of the paper, the transfer-function framework is applied to the study of feedback control. New results are obtained on assignability of system dynamics by using dynamic output feedback and dynamic state feedback. The results are applied to the control of an armature-controlled do motor with a variable loading.

A nonconnective delooping of algebraic K-theory

Abstract: Given a ring R, it is known that the topological space BGl(R)+ is an infinite loop space. One way to construct an infinite loop structure is to consider the category F of free R-modules, or rather its classifying space BF, as food for suitable infinite loop space machines. These machines produce connective spectra whose zeroth space is (BF)+ = ZXBG1(R)+. In this paper we consider categories Co(F) = F, C1(F),... of parametrized free modules and bounded homomorphisms and show that the spaces (BCo)+ = (BF)+, (BC1)+,... are the connected components of a nonconnective ω-spectrum BC(F) with πiBC(F) = Ki(R) even for negative i.

The Krull-Gabriel dimension of the representation theory of a tame hereditary Artin algebra and applications to the structure of exact sequences

Abstract: For an Artin Algebra Λ of finite representation type, the category Λ-mod, considered as a ring with several objects, has Krull dimension zero. Contrary, for a wild hereditary Artin Algebra this dimension does not exist. In this paper we show that the Krull dimension of Λ-mod for an Artin Algebra Λ of tame representation type is two. The corresponding Krull-Gabriel filtration by Serre subcategories of the category F of finitely presented contravariant functors on Λ-mod leads to a hierarchy of exact sequences in Λ-mod. Influenced by the functorial approach to almost split sequences by M. Auslander and I. Reiten, we investigate the exact sequences whose corresponding functors become simple in one of the successive quotient categories of the Krull-Gabriel filtration of F.

Short-Packet Transfer Performance in Local Area Ring Networks

Abstract: The performance of various access control protocols for bit-serial local area computer network (LACN) rings is studied. Applications in which mnessage packets are of fixed length and shorter than the total inherent propagation delay around the ring are the focus of attention. Token, slotted, and static and dynamic insertion rings are included in the study. In all cases, the transmitting station is responsible for removing its transmitted packet from the ring. Under this type of removal rule, it is possible for the stations of all types of ring structures to execute their access control algorithms with only a short fixed in-line delay in each station. The insertion rings dynamically switch longer delays (insertion registers) into the ring when they are transmitting a packet. The transmitter-remove rule operates in such a way that hogging of the ring transmission facility by a subset of stations cannot occur. Expressions that approximate average transfer time as a function of utilization are derived for all ring types and are checked by simulation. The expressions are found to be quite accurate at low ring utilization levels, which is the case of most importance for LACN's; but some of them exhibit significant errors at high utilization levels. For the assumed, short-packet environment, it is found that token rings exhibit the slowest transfer times, while dynamic insertion rings are fastest.

On Hilbert's 17th problem and real Nullstellensatz for global analytic functions

Abstract: The author proves a Nullstellensatz for the ring of real analytic functions on a compact analytic manifold. The main results are the following. Theorem 1: Let X be a compact irreducible analytic set of a real analytic manifold M and f:X→R a nonnegative analytic function. Then f is a sum of squares of meromorphic functions. Theorem 2: Let I be a finitely generated ideal of O(M) with Z(I) compact. Then IZ(I)=I√R, where Z(I) is the zero set of I, IZ(I) the ideal of functions (in O(M)) vanishing on I, and I√R the real radical of I (i.e. the set of functions f in O(M) such that there exist g1,⋯,gk and an integer p with f2p + g2 1 + ⋯ +g2k ∈ I). Corollary: Let I be as in Theorem 2. Then IZ(I)=I if and only if I is real (i.e. I=I √ R). The proofs are based on results about extension of orders.

Doubly Linked Ring Networks

Abstract: We consider networks of processors where each processor either has one in-link and one out-link, or two in-links and two out-links. We study three properties of such networks: 1) diameter, 2) connectivity, and 3) the ring property. We propose a class of networks which seem to achieve the optimum as far as these three properties are concerned.

Diophantine equations, Hilbert series, and undecidable spaces

Abstract: Summary In 1979 Jan-Erik Roos published a fascinating paper [22] making explicit connections between the Poincare-Betti series of loop spaces or local rings and the Hilbert series of finitely presented graded Hopf algebras. This paper develops a way to model Diophantine equations within the category of graded Hopf algebras. Combining this with Roos' work, we obtain loop spaces and local rings whose series reflect the solution sets of arbitrary Diophantine equations. By Matiyasevic's negative solution to Hilbert's tenth problem ([15], [16]), there is no algorithm for deciding in general whether or not a given Diophantine equation has solutions. For us, one consequence is that no algorithm exists to decide whether a given finitely presented graded algebra is "generic" (see [3]). As to rings, there is no finite procedure for evaluating whether an arbitrary ring's Poincare series equals a given series, even though by [10] the sequence of coefficients is recursive. Likewise, given two finite simply-connected CW complexes, there is no guaranteed method to tell, in general, whether their loop spaces have the same rational homotopy type. At the same time, Matiyasevic showed that there exist undecidable Diophantine equations, i.e., polynomial equations for which neither the existence nor the non-existence of integer solutions may be proved in a given axiomatization of arithmetic. We obtain "undecidable local rings," "undecidable spaces," and "undecidable topological maps." These concepts will be made precise in Section 4. A few other consequences of the link between Diophantine equations and Hilbert series may be listed. First, a graded algebra (resp. loop space or local ring) exists whose Hilbert (resp. Poincare) series radius of convergence may be proved to be a transcendental number. Also, the Hilbert series can represent a transcendental function which solves no algebraic differential equation. Lastly, we prove a stability theorem for "Diophantine solution sets of bounded complexity."

The Cartan matrix as an indicator of finite global dimension for Artinian rings

Abstract: A left serial ring has finite global dimension if and only if its Cartan matrix has determinant equal to 1.

Equational theory of positive numbers with exponentiation

Abstract: A. Tarski asked if all true identities involving 1, addition, multiplication, and exponentiation can be derived from certain so-called "high-school" identities (and a number of related questions). I prove that equational theory of (N, 1, +, , T) is decidable (a T b means ac' for positive a, b) and that entailment relation in this theory is decidable (and present a similar result for inequalities). A. J. Wilkie found an identity not derivable from Tarski's axioms with a difficult proof-theoretic argument of nonderivability. I present a model of Tarski's axioms consisting of 59 elements in which Wilkie's identity fails. 1. This note is related to "Tarski's high school algebra problem" and a number of other model-theoretic questions concerning exponentiation of positive real numbers and positive integers (see e.g. [1]). Let a T b = a b for positive a, b, and L = the set of terms in signature (1, +, *, T). As always, R+ is the set of positive reals. We give proofs of decidability for two problems about identities, and we also present a 59-element model in which Tarski's "high school algebra" identities are true, while Wilkie's identity is false. Our first result gives a new proof of a theorem of A. Macintyre [3] (proved for terms in one variable by Richardson [4]). THEOREM 1. Let X be any subset of R+ containing 1 and closed under addition, multiplication, and exponentiation. Then the set of valid equalities T = { t1 = t2l tl, t2 E L, X W= t1 = t2 } is decidable and does not depend on X. The proof is based on the following lemma, which is proved in ??2 and 3. LEMMA 1. There is a recursive function M: L X L -N such that, for any t1(r, s), t2(r, s) E L, for any positive real (values of ) -Fif card{s E R+ It1(r, s) = t2Qr, s)} > M(t1, t2) then Vs E R+: t1(r, s) = t2(r, s). PROOF OF THEOREM 1. Proceed by induction on the number of variables: Vs E X: t1(r, s) = t2(r, s) is equivalent to &m 1t1(r, k) = t2(r-, k), where M = M(t1, t2). Received by the editors February 4, 1983 and, in revised form, April 2, 1984. 1980 Mathematics Subject Classification. Primary 03B25, 03C05, 03C13. Kev words and phrases. Exponentiation of positive reals, exponentiation of positive integers, Tarski's high school algebra problem, decidability of equational theory, decidability of entailment relation, differential ring, finite model of Tarski's axioms. ?01985 American Mathematical Society 0002-9939/85 $1.00 + $.25 per page

Topics in Geometric Algebra over Rings

Abstract: E. ARTIN (1957), R. BAER (1952) and J.DIEUDONNE (1951) emphasized many times the structural identity between classical projective geometry and linear algebra over a division ring. Then Baer pointed out a possible extension of this structural identity to the case of a ring, generating intense research activity in the area of geometric algebra over rings. In this direction the most significant results appear: Ojanguren and Sridharan’s article on the fundamental theorem of projective geometry over a commutative ring; some theorems due to Klingenberg, Bass and Suslin on the structure of the general linear group over appropriate rings and some valuable notes of O’Meara’s on the automorphisms of linear groups over an integral domain.

Isomorphisms of unitary groups over associative rings

Abstract: Automorphisms of unitary linear groups u (n,Λ,q) over the ring Λ are found to be standard on an elementary unitary subgroup in the case when the hyperbolic rank of the form q is strictly greater than one and n⩾5 (for a commutative ring, n⩾4).

On $G$-systems and $G$-graded rings

Abstract: Rings graded by finite groups and homomorphic images of such rings are studied. Obtained results concern finiteness conditions and radicals. Introduction. Our aim in this paper is the study of rings graded by finite groups. To obtain some results we need information on homomorphic image of a graded ring (cf. Theorem 5). The proofs of other results work in this more general situation as well. For these reasons the paper concerns G-systems, defined as follows. Let G be a finite group with identity e. A ring R is said to be G-system if R = EgEGRgR where Rg are such additive subgroups of R that RgRh c Rgh for all g, h E G. If for all g, h E G, RgRh = Rgh, R is called [3,4,7] the Clifford system. Certainly any G-graded ring is a G-system and the class of G-systems is homomorphically closed, while G-graded rings do not necessarily have this property. It is easy to check that every G-system is a homomorphic image of a G-graded ring. In this paper we prove that a "Clifford type" theorem holds for every G-system R. Namely, we show that simple R-modules (R-bimodules) are completely reducible Re-modules (Re-bimodules) and that J(Re) = J(R) r) Re, U(Re) C U(R), where J(-), U(-) denote the Jacobson and the Brown-McCoy radical, respectively. Using this method we obtain, in particular, a quite different proof from those given by M. Cohen and S. Montgomery [2] and M. Van den Bergh [6] of Bergman's conjecture. We also prove that the R-module M is Noetherian if and only if M is Noetherian as an R e-module.

Decision Problems Concerning $S$-Arithmetic Groups

Abstract: This paper is a continuation of our previous work in [12]. The results, and some applications, have been described in the announcement [13]; it may be useful to discuss here, a little more fully, the nature and purpose of this work. We are concerned basically with three kinds of algorithmic problem: (1) isomorphism problems, (2) “orbit problems”, and (3) “effective generation”. (1) Isomorphism problems . Here we have a class of algebraic objects of some kind, and ask: is there a uniform algorithm for deciding whether two arbitrary members of are isomorphic? In most cases, the answer is no: no such algorithm exists. Indeed this has been one of the most notable applications of methods of mathematical logic in algebra (see [26, Chapter IV, §4] for the case where is the class of all finitely presented groups). It turns out, however, that when consists of objects which are in a certain sense “finite-dimensional”, then the isomorphism problem is indeed algorithmically soluble. We gave such algorithms in [12] for the following cases: = {finitely generated nilpotent groups}; = {(not necessarily associative) rings whose additive group is finitely generated}; = {finitely Z -generated modules over a fixed finitely generated ring}. Combining the methods of [12] with his own earlier work, Sarkisian has obtained analogous results with the integers replaced by the rationals: in [20] and [21] he solves the isomorphism problem for radicable torsion-free nilpotent groups of finite rank and for finite-dimensional Q -algebras.

Iterations and logarithms of formal automorphisms

Abstract: Using the decomposition of an automorphism of the ring of formal power series in several variables, in a semisimple and a unipotent automorphism, I prove in this paper that an automorphism allows a continuous iteration if and only if it is the exponential of a derivation. This result implies a number of results recently obtained by Reich, Schwaiger, and Bucher.

Rings with flat right ideals and distributive rings

Abstract: RINGS WITH FLAT RIGHT IDEALS AND DISTRIBUTIVE RINGS A. A. Tuganbaev All rings are assumed to be associative with nonzero unity, all modules are unitary and unless the side is specified, right. A module with a distributive lattice of submodules is called a distributive module. Expressions like "a distributive ring" mean that corre- sponding conditions are satisfied both on the right and on the left. Dedekind rings (for in- stance, the ring of integers) are distributive. The inequality w.gl.dim.(R)~ t, where w,gl.dim (R) is the weak global dimension of the ring R, means that all right (left) ideals of the ring R are flat. Jensen [I] has proved for a commutative semiprime ring R that the inequality w.gl.dim (R)~I is equivalent to distributivity of the ring R. The main result of the present note is THEOREM i. For a semiprime ring R integralover its center the following conditions are equivalent: (a) the ring R is distributive; (b) w.gl.dim (R)~I, and the set R\P is an Ore set for each prime ideal P of the ring

On commutative ring spectra of characteristic 2

Abstract: The following theorem (a conjecture of Rourke) is proved: every commutative ring spectrum of characteristic 2 which has finite type is isomorphic to the spectrum of ordinary cohomology theory with coefficients in .Bibliography: 13 titles.

The functional equation for Poincaré series of trace rings of generic 2×2 matrices

Abstract: In this note we give a rational expression for the Poincare series of Πm,2, the trace ring ofm generic 2×2 matrices. This result extends the computations of E. Formanek form⩽4. As a consequence, we prove that the Poincare series satisfies the functional equation(IIm,2;1/t)=-t4m.P(IIm,2,t) (m>2) supporting the conjecture that Πm,2 is a Gorenstein ring.