Showing papers on "Ring (mathematics) published in 1995"

Reasoning about rings

Abstract: The ring is a useful means of structuring concurrent processes. Processes communicate by passing a token in a fixed direction; the process that possesses the token is allowed to make certain moves. Usually, correctness properties are expected to hold irrespective of the size of the ring. We show that the problem of checking many useful correctness properties for rings of all sizes can be reduced to checking them on a ring of small size. The results do not depend on the processes being finite state. We illustrate our results on examples.

Synthesis of Mu1 tiple-Ring -Re sonator Filters for Optical Systems

Abstract: A method for the synthesis of optical filters consisting of a cascade of N-coupled rings is presented. The procedure is based on the definition of a polynomial whose roots are the zeros of the channel-dropping transmittance characteristic and provides directly the ring electrical lengths and the mutual coupling coefficients. A design example of a Chebyshev-type 6- ring bandpass filter is presented.

The Disturbance Decoupling Problem for Systems Over a Ring

Abstract: Up to now the use of geometric methods in the study of disturbance decoupling problems (DDPs) for systems over a ring has provided only necessary conditions for the existence of solutions. In this paper we study such problems, considering separately the case in which only static feedback solutions are allowed, and the one in which dynamic feedback solutions are admitted. In the first case, we give a complete geometric characterization of the solvability conditions of such problems for injective systems with coefficients in a commutative ring. Practical procedures for testing the solvability conditions and for constructing solutions, if any exist, are given in the case of systems with coefficients in a principal ideal domain (PID). In the second case, we give a complete geometric characterization of the solvability conditions for systems with coefficients in a PID.

Test ideals in local rings

Abstract: It is shown that certain aspects of the theory of tight closure are well behaved under localization. Let J be the parameter test ideal for R, a complete local Cohen-Macaulay ring of positive prime characteristic. For any multiplicative system U c R, it is shown that JU -1R is the parameter test ideal for U-1 R. This is proved by proving more general localization results for the here-introduced classes of "F-ideals" of R and "F-submodules of the canonical module" of R, which are annihilators of R modules with an action of Frobenius. It also follows that the parameter test ideal cannot be contained in any parameter ideal of R. Tight closure has produced surprising new results (for instance, that the absolute integral closure of a complete local domain R of prime characteristic is a Cohen-Macaulay algebra for R) as well as tremendously simple proofs for otherwise difficult theorems (such as the fact that the ring of invariants of a linearly reductive group acting on a regular ring is Cohen-Macaulay). The definition of tight closure is recalled in Section 1, but we refer the reader to the papers of Hochster and Huneke listed in the bibliography for more about its applications. While primarily a prime characteristic notion, tight closure offers insight into arbitrary commutative rings containing Q by fairly standard "reduction to characteristic p" techniques. Despite its successes, the tight closure operation, which in its principal setting is a closure operation performed on ideals in a commutative Noetherian ring of prime characteristic, remains poorly understood. For example, tight closure is not known to behave well under localization; given an ideal I of a ring R, letting I* denote the tight closure of I, it is not known whether or not

SDH ring high order path management

Abstract: A method of managing a telecommunication subnetwork system composed of a number of network elements coupled together through an optical medium to form a Synchronous Digital Hierarchy (SDH) network ring. Each of the network elements receives upon initiation or reconfiguration and then stores an identical ring table or knowledge database that is composed of data in a predetermined format and structure. The ring table defines the ring type and ring identification, number of nodes, ring status, node identification, sequence, and status, and provides ring provisioning tables and embedded SDH ring path identification/status information. The ring table is utilized in each of the network elements for provisioning and other management type functions common within a SDH ring environment.

Isomorphisms of standard operator algebras

Abstract: Let X and Y be Banach spaces, dim X = oo, and let v and g be standard operator algebras on X and Y, respectively. Assume that q$: vW -q is abijective mapping satisfying II0(AB)-0(A)0(B)II < e, A, B E , where e is a given positive real number (no linearity or continuity of q is assumed). Then q is a spatially implemented linear or conjugate linear algebra isomorphism. In particular, q is continuous. Let X be a Banach space. By R(X) we mean the algebra of all bounded linear operators on X. We denote by F(X) the subalgebra of bounded finite rank operators. We shall call a subalgebra v of R(X) standard provided v contains F(X) (X need not be closed). For any x E X and f E X' we denote by x? f the bounded linear operator on X defined by (x? f)y = f(y)x for y E X. Note that every operator of rank one can be written in this form. The operator x 0 f is a projection if and only if f(x) = 1 . Let X and Y be Banach spaces, and let v and 7 be standard operator algebras on X and Y, respectively. It is a classical result [4] that every algebra isomorphism q: v 7 is spatial, i.e., there is a linear topological isomorphism T: X -Y such that q(A) = TAT-1 for all A E ,v. When discussing isomorphisms of algebras one usually assumes that these mappings are linear. A more general approach is to consider the algebra only as a ring. It seems that the first step in this direction was made by Rickart [9, Theorem 3.2], who treated isomorphisms of primitive real Banach algebras which are not assumed to be linear, i.e., they are isomorphisms merely in the ring sense. The famous result of Kaplansky [6, 7] decomposes a ring isomorphism between two semisimple complex Banach algebras into a linear part, a conjugate linear part, and a nonreal linear part on a finite-dimensional ideal. Let R be a ring. Recall that R is called prime if aRb = 0 implies a = 0 or b = 0. Assume that a prime ring R contains an idempotent e # 0, 1 (R need not have an identity). Then every multiplicative bijective mapping of R onto an arbitrary ring S is additive [8]. It is an easy consequence of the Hahn-Banach theorem that R(X) is a prime ring. Thus, the above-mentioned results imply that if dimX = ox, then every multiplicative bijective mapping q of R (X) onto R (Y) is of the form +(A) = TAT-1 , where T: X -Y is Received by the editors May 11, 1993 and, in revised form, October 18, 1993. 1991 Mathematics Subject Classification. Primary 47D30. Supported by a grant from the Ministry of Science of Slovenia. ? 1995 American Mathematical Society 0002-9939/95 $1.00 + $.25 per page

Stable range one for rings with many idempotents

Abstract: An associative ring R is said to have stable range 1 if for any a, b e R satisfying aR + bR = R, there exists y e R such that a + by is a unit. The purpose of this note is to prove the following facts. Theorem 3: An exchange ring R has stable range 1 if and only if every regular element of R is unit-regular. Theorem 5: If R is a strongly 7r-regular ring with the property that all powers of every regular element are regular, then R has stable range 1. The latter generalizes a recent result of Goodearl and Menal [5]. Let R be an associative ring with identity. R is said to have stable range 1 if for any a, b E R satisfying aR + bR = R, there exists y E R such that a + by is a unit. This definition is left-right symmetric by Vaserstein [9, Theorem 2]. Furthermore, by a theorem of Kaplansky, all one-sided units are two-sided in rings having stable range 1 (cf. Vaserstein [10, Theorem 2.6]). It is well known that a (von Neumann) regular ring R has stable range 1 if and only if R is unit-regular (see, for example, Goodearl [4, Proposition 4.12]). Call a ring R strongly 7-regular if for every element a E R there exist a number n (depending on a) and an element x E R such that an = an+lx. This is in fact a two-sided condition [3]. It is an open question whether all strongly 7r-regular rings have stable range 1. Goodearl and Menal [5] proved that strongly 7r-regular rings are unit-regular and, hence, have stable range 1 (Theorem 5.8, p. 278). In this note we first extend the above result for von Neumann regular rings to a larger class of rings, which includes all strongly 7r-regular rings, 7r-regular rings, von Neumann regular rings, and algebraic algebras. As an application of this, we prove that a strongly 7r-regular ring R has stable range 1 if powers of every regular element are regular. The latter is a generalization of the abovementioned result of Goodearl and Menal for strongly 7r-regular regular rings. As one can see from our proofs, rings in these classes have a large supply of idempotents. Throughout, R stands for an associative ring with identity and J(R) for the Jacobson radical of R. Modules are unitary right R-modules except otherwise specified. For other undefined terms, readers are referred to [4]. Let MR be a right R-module. Following Crawley and Jonsson [2], MR is said to have the exchange property if for every module AR and any two Received by the editors January 24, 1994 and, in revised form, May 3, 1994; originally communicated to the Proceedings of the AMS by K. A. Goodearl. 1991 Mathematics Subject Classification. Primary 1 6D70, 1 6P70.

Quantum cohomology of flag varieties

Abstract: We describe a construction of Gromov-Witten invariants for flag varieties and use it to give a presentation for the quantum cohomology ring, by extending the ideas used by Bertram in the case of Grassmannians. This provides a proof for the conjecture of Givental and Kim in [GK].

Evolutions, Symbolic Squares, and Fitting Ideals

Abstract: Given a reduced local algebra T over a suitable ring or field k we study the question of whether there are nontrivial algebra surjections R → T which induce isomorphisms R/k ⊗ T → T/k . Such maps, called evolutions, arise naturally in the study of Hecke algebras, as they implicitly do in the recent work of Wiles, Taylor-Wiles, and Flach. We show that the existence of non-trivial evolutions of an algebra T can be characterized in terms of the symbolic square of an ideal defining T. We give a characterization of the symbolic square in terms of Fitting ideals. Using this and other techniques we show that certain classes of reduced algebras — codimension 2 Cohen-Macaulay, Codimension 3 Gorenstein, licci algebras in general, and some others — admit no nontrivial evolutions. On the other hand we give examples showing that non-trivial evolutions of reduced CohenMacaulay algebras of codimension 3 do exist in every positive characteristic.

Undergraduate commutative algebra

Abstract: Commutative algebra is at the crossroads of algebra, number theory and algebraic geometry. This textbook is affordable and clearly illustrated, and is intended for advanced undergraduate or beginning graduate students with some previous experience of rings and fields. Alongside standard algebraic notions such as generators of modules and the ascending chain condition, the book develops in detail the geometric view of a commutative ring as the ring of functions on a space. The starting point is the Nullstellensatz, which provides a close link between the geometry of a variety V and the algebra of its coordinate ring A=k[V]; however, many of the geometric ideas arising from varieties apply also to fairly general rings. The final chapter relates the material of the book to more advanced topics in commutative algebra and algebraic geometry. It includes an account of some famous 'pathological' examples of Akizuki and Nagata, and a brief but thought-provoking essay on the changing position of abstract algebra in today's world.

Failure restoration method in a mesh network

Abstract: A method for constructing a network capable of self-healing from failure in a mesh network and a restoration algorithm are provided. A logical ring is set for each closed loop in the network, and when a failure occurs, the affected traffic is re-routed in the each logical ring toward the opposite direction away from the failure to restore the failed network. The network is divided into a plurality of logical rings to establish the restoration route so that time required for the restoration is shortened.

Broadcast ring sandwich networks

Abstract: In this paper we present a constructive design of a new class of cascaded network structures for broadcast applications called ring sandwich networks. These ring sandwich networks are rearrangeable in the sense that a request for a connection between a sender and a receiver can sometimes be realized only by first rearranging other existing connection paths through the network. We present analytical results which permit the average rearrangeability of ring sandwich networks to be evaluated on the basis of fundamental structural parameters associated with the ring sandwich design so that the trade-off between the network rearrangeability and the network cost can be determined. It is shown that the average number of rearrangements to satisfy a broadcast connection request relative to the subnetwork of the cascaded ring sandwich structure providing fanout can be reduced to O(1); this is in contrast to O(N) for other existing cascaded designs. We give detailed connecting algorithms that can be used to satisfy connection requests. We also support our analytically derived results with corroborating simulation data. This work provides an analytical framework for a class of low-cost broadcast networks currently being employed by government and industry in both broadcasting and conferencing applications wherein only a limited degree of rearrangements can be tolerated. >

Quantum cohomology of flag varieties

Abstract: We describe a construction of Gromov-Witten invariants for flag varieties and use it to give a presentation for the quantum cohomology ring, by extending the ideas used by Bertram in the case of Grassmannians. This provides a proof for the conjecture of Givental and Kim in [GK].

The (0,2) Exactly Solvable Structure of Chiral Rings, Landau-Ginzburg Theories and Calabi-Yau Manifolds

Abstract: We identify the exactly solvable theory of the conformal fixed point of (0,2) Calabi-Yau sigma-models and their Landau-Ginzburg phases. To this end we consider a number of (0,2) models constructed from a particular (2,2) exactly solvable theory via the method of simple currents. In order to establish the relation between exactly solvable (0,2) vacua of the heterotic string, (0,2) Landau-Ginzburg orbifolds, and (0,2) Calabi-Yau manifolds, we compute the Yukawa couplings in the exactly solvable model and compare the results with the product structure of the chiral ring which we extract from the structure of the massless spectrum of the exact theory. We find complete agreement between the two up to a finite number of renormalizations. For a particularly simple example we furthermore derive the generating ideal of the chiral ring from a (0,2) linear sigma-model which has both a Landau-Ginzburg and a (0,2) Calabi-Yau phase.

Linear Stability Analysis of Some Symmetrical Classes of Relative Equilibria

Abstract: The linear stability of several classes of symmetrical relative equilibria of the Newtonian n-body problem are studied. Most turn out to be unstable; however, a ring of at least seven small equal masses around a sufficiently large central mass is stable.

Modules with semi-local endomorphism ring

Abstract: We use the concept of dual Goldie dimension and a characteriza- tion of semi-local rings due to Camps and Dicks (1993) to find some classes of modules with semi-local endomorphism ring. We deduce that linearly com- pact modules have semi-local endomorphism ring, cancel from direct sums and satisfy the n th root uniqueness property. We also deduce that modules over commutative rings satisfying AB5* also cancel from direct sums and satisfy the n th root uniqueness property. Let R be an associative ring with 1 and let Af be a right unital i?-module. A finite set Ax, ... , An of proper submodules of M is said to be coindependent if for each /, 1 < i < n, Ai + f\j:jii Aj = M, and a family of submodules of M is said to be coindependent if each of its finite subfamilies is coindependent. The module M is said to have finite dual Goldie dimension if every coindependent family of submodules of M is finite. It can be shown that, in this case, there is a maximal coindependent family of submodules of Af. If this set is finite, then its cardinality (denoted by codim(Af) ) is uniquely determined and is called the dual Goldie dimension of Af. If this set is infinite we set codim(Af ) = oo and say that Af has infinite dual Goldie dimension. A module with dual Goldie dimension 1 is said to be hollow, and a cyclic hollow module is said to be local. We have

Evolutions, Symbolic Squares, and Fitting Ideals

Abstract: Given a reduced local algebra $T$ over a suitable ring or field $k$ we study the question of whether there are nontrivial algebra surjections $R\to T$ which induce isomorphisms $\Omega_{R/k}\otimes T \to \Omega_{T/k}$. Such maps, called evolutions, arise naturally in the study of Hecke algebras, as they implicitly do in the recent work of Wiles, Taylor-Wiles, and Flach. We show that the existence of non-trivial evolutions of an algebra $T$ can be characterized in terms of the symbolic square of an ideal defining $T$. We give a characterization of the symbolic square in terms of Fitting ideals. Using this and other techniques we show that certain classes of reduced algebras -- codimension 2 Cohen-Macaulay, Codimension 3 Gorenstein, licci algebras in general, and some others -- admit no nontrivial evolutions. On the other hand we give examples showing that non-trivial evolutions of reduced Cohen-Macaulay algebras of codimension 3 do exist in every positive characteristic.

Left annihilators characterized by GPIs

Abstract: Let R be a semiprime ring with extended centroid C, U the right Utumi quotient ring of R, S a subring of U containing R and PI, P2 two right ideals of R. In the paper we show that 1S(Pi) = 1S(P2) if and only if Pi and P2 satisfy the same generalized polynomial identities (GPIs) with coefficients in SC, where lS(pi) denotes the left annihilator of pi in S. As a consequence of the result, if p is a right ideal of R such that IR(P) = 0, then p and U satisfy the same GPIs with coefficients in the two-sided Utumi quotient ring of R. This paper is motivated by Chuang's paper [3] and Beidar's paper [2]. Recall that a ring R is said to be a left faithful ring if, for a E R, aR = 0 implies a = 0. For a left faithful ring R, the right Utumi quotient ring of R can be characterized as a ring U satisfying the following axioms: (1) R is a subring of U. (2) For each a E U, there exists a dense right ideal p of R such that ap C R. (3) If a E U and ap = 0 for some dense right ideal p of R, then a = 0. (4) For any dense right ideal p of R and for any right R-module homomorphism q : PR -RR, there exists a E U such that q(x) = ax for all X E P. Let R be a left faithful ring and p be a dense right ideal of R. We note that p itself is a left faithful ring. Furthermore, p and R have the same right Utumi quotient ring. More precisely, denote by U(R) (U(p) resp.) the right Utumi quotient ring of R (p resp.). Then there exists a ring isomorphism h from U(p) onto U(R) such that h(x) = x for all x E p . In [3] Chuang proved the theorem: Let R be a prime ring, U its right Utumi quotient ring and NR a dense R-submodule of UR. Then N and U satisfy the same generalized polynomial identities (GPIs) with coefficients in U. In this theorem we note that N n R is always a dense right ideal of R. Since N n R and R have the same right Utumi quotient ring, Chuang's theorem just says that R and U satisfy the same GPIs with coefficients in U. Also, in an earlier paper [2] Beidar proved that the same result remains true for semiprime rings. For a semiprime ring R we observe that N n R is a dense right ideal of R for any dense R-submodule NR of UR . Also, lu (N n R), the left annihilator of N n R Received by the editors April 11, 1994 and, in revised form, July 8, 1994; originally communicated to the Proceedings of the AMS by Ken Goodearl. 1991 Mathematics Subject Classification. Primary 1 6R50.

A Ring Median Filter for Digital Images

Abstract: The ring filter is defined as a median filter which assigns weight only to select pixels in an annulus. Its advantage is that it has a sharply-defined scale length; that is, all objects with a scale-size less than the radius of the ring are filtered and replaced by the local background level. It provides a fast, simple and intuitive method to remove the small-scale objects (independent of morphology) from a digital image, leaving behind the large-scale objects and overall light gradients. The ring filter is much faster than the more commonly used filled-box median filter, and completes in one or two passes what previously required a long iterative procedure. Several examples of its use are presented.

Centralizer near-rings that are rings

Abstract: Given an R-module M, the centralizer near-ring ℳR (M) is the set of all functions f: M → M with f(xr)= f(x)r for all x ∈ M and r∈R endowed with point-wise addition and composition of functions as multiplication. In general, ℳR(M) is not a ring but is a near-ring containing the endomorphism ring ER(M) of M. Necessary and/or sufficient conditions are derived for ℳR(M) to be a ring. For the case that R is a Dedekind domain, the R-modules M are characterized for which (i) ℳR(M) is a ring; and (ii)ℳR(M) = ER(M). It is shown that over Dedekind domains with finite prime spectrum properties (i) and (ii) are equivalent.

Rings whose modules have maximal submodules

Abstract: A ring $R$ is a right max ring if every right module $M eq 0$ has at least one maximal submodule. It suffices to check for maximal submodules of a single module and its submodules in order to test for a max ring; namely, any cogenerating module $E$ of $\operatorname{mod}$-$R$; also it suffices to check the submodules of the injective hull $E(V)$ of each simple module $V$ (Theorem 1). Another test is transfinite nilpotence of the radical of $E$ in the sense that $\operatorname{rad}^{\alpha}E=0$; equivalently, there is an ordinal $\alpha$ such that $\operatorname{rad}^{\alpha}(E(V))=0$ for each simple module $V$. This holds iff each $\operatorname{rad}^{\beta}(E(V))$ has a maximal submodule, or is zero (Theorem 2). If follows that $R$ is right max iff every nonzero (subdirectly irreducible) quasi-injective right $R$-module has a maximal submodule (Theorem 3.3). We characterize a right max ring $R$ via the endomorphism ring $\Lambda$ of any injective cogenerator $E$ of $\operatorname{mod}$-$R$; namely, $\Lambda/L$ has a minimal submodule for any left ideal $L=\operatorname{ann}_{\Lambda}M$ for a submodule (or subset) $M e 0$ of $E$ (Theorem 8.8). Then $\Lambda/L_0$ has socle $ e 0$ for: (1) any finitely generated left ideal $L_0 e\Lambda$; (2) each annihilator left ideal $L_0 e \Lambda$; and (3) each proper left ideal $L_0=L+L'$, where $L=\operatorname{ann}_{\Lambda}M$ as above (e.g. as in (2)) and $L'$ finitely generated (Corollary 8.9A).