Showing papers on "Ring (mathematics) published in 1998"

Symmetric spectra

Abstract: The long hunt for a symmetric monoidal category of spectra finally ended in success with the simultaneous discovery of the third author's discovery of symmetric spectra and the Elmendorf-Kriz-Mandell-May category of S-modules. In this paper we define and study the model category of symmetric spectra, based on simplicial sets and topological spaces. We prove that the category of symmetric spectra is closed symmetric monoidal and that the symmetric monoidal structure is compatible with the model structure. We prove that the model category of symmetric spectra is Quillen equivalent to Bousfield and Friedlander's category of spectra. We show that the monoidal axiom holds, so that we get model categories of ring spectra and modules over a given ring spectrum.

Rings of quotients

Abstract: Let T be a set of elements in a ring A. The set T is right permutable if for any a ∈ A and t ∈ T, there exist b ∈ A, u ∈ T such that au = tb. A multiplicative set in a ring A is any subset T of A such that 1 ∈ T,0 ∉ T and T is closed under multiplication. A completely prime ideal in a ring A is any proper ideal B such that A\B is a multiplicative set (i.e. A/Bis a domain). A minimal prime ideal (resp. minimal completely prime ideal) in a ring A is any prime (resp. completely prime) ideal P such that P contains no properly any other prime ideal (resp. completely prime ideal) of A. Let I be any proper ideal of a ring A. The set of all elements a ∈ A such that a + I is a regular element of A/I is denoted by c(I). In particular, c(0) is the set of all regular elements of A.

Discrete and Combinatorial Mathematics: An Applied Introduction

Abstract: PART 1. FUNDAMENTALS OF DISCRETE MATHEMATICS. 1. Fundamental Principles of Counting. The Rules of Sum and Product. Permutations. Combinations: The Binomial Theorem. Combinations with Repetition. The Catalan Numbers (Optional). Summary and Historical Review. 2. Fundamentals of Logic. Basic Connectives and Truth Tables. Logical Equivalence: The Laws of Logic. Logical Implication: Rules of Inference. The Use of Quantifiers. Quantifiers, Definitions, and the Proofs of Theorems. Summary and Historical Review. 3. Set Theory. Sets and Subsets. Set Operations and the Laws of Set Theory. Counting and Venn Diagrams. A First Word on Probability. The Axioms of Probability (Optional). Conditional Probability: Independence (Optional). Discrete Random Variables (Optional). Summary and Historical Review. 4. Properties of the Integers: Mathematical Induction. The Well-Ordering Principle: Mathematical Induction. Recursive Definitions. The Division Algorithm: Prime Numbers. The Greatest Common Divisor: The Euclidean Algorithm. The Fundamental Theorem of Arithmetic. Summary and Historical Review. 5. Relations and Functions. Cartesian Products and Relations. Functions: Plain and One-to-One. Onto Functions: Stirling Numbers of the Second Kind. Special Functions. The Pigeonhole Principle. Function Composition and Inverse Functions. Computational Complexity. Analysis of Algorithms. Summary and Historical Review. 6. Languages: Finite State Machines. Language: The Set Theory of Strings. Finite State Machines: A First Encounter. Finite State Machines: A Second Encounter. Summary and Historical Review. 7. Relations: The Second Time Around. Relations Revisited: Properties of Relations. Computer Recognition: Zero-One Matrices and Directed Graphs. Partial Orders: Hasse Diagrams. Equivalence Relations and Partitions. Finite State Machines: The Minimization Process. Summary and Historical Review. PART 2. FURTHER TOPICS IN ENUMERATION. 8. The Principle of Inclusion and Exclusion. The Principle of Inclusion and Exclusion. Generalizations of the Principle. Derangements: Nothing Is in Its Right Place. Rook Polynomials. Arrangements with Forbidden Positions. Summary and Historical Review. 9. Generating Functions. Introductory Examples. Definition and Examples: Calculational Techniques. Partitions of Integers. The Exponential Generating Functions. The Summation Operator. Summary and Historical Review. 10. Recurrence Relations. The First-Order Linear Recurrence Relation. The Second-Order Linear Homogeneous Recurrence Relation with Constant Coefficients. The Nonhomogeneous Recurrence Relation. The Method of Generating Functions. A Special Kind of Nonlinear Recurrence Relation (Optional). Divide and Conquer Algorithms. Summary and Historical Review. PART 3. GRAPH THEORY AND APPLICATIONS. 11. An Introduction to Graph Theory. Definitions and Examples. Subgraphs, Complements, and Graph Isomorphism. Vertex Degree: Euler Trails and Circuits. Planar Graphs. Hamilton Paths and Cycles. Graph Coloring and Chromatic Polynomials. Summary and Historical Review. 12. Trees. Definitions, Properties, and Examples. Rooted Trees. Trees and Sorting. Weighted Trees and Prefix Codes. Biconnected Components and Articulation Points. Summary and Historical Review. 13. Optimization and Matching. Dijkstra's Shortest Path Algorithm. Minimal Spanning Trees: The Algorithms of Kruskal and Prim. Transport Networks: The Max-Flow Min-Cut Theorem. Matching Theory. Summary and Historical Review. PART 4. MODERN APPLIED ALGEBRA. 14. Rings and Modular Arithmetic. The Ring Structure: Definition and Examples. Ring Properties and Substructures. The Integers Modulo n. Cryptology. Ring Homomorphisms and Isomorphisms: The Chinese Remainder Theorem. Summary and Historical Review. 15. Boolean Algebra and Switching Functions. Switching Functions: Disjunctive and Conjunctive Normal Forms. Gating Networks: Minimal Sums of Products: Karnaugh Maps. Further Applications: Don't-Care Conditions. The Structure of a Boolean Algebra (Optional). Summary and Historical Review. 16. Groups, Coding Theory, and Polya's Theory of Enumeration. Definition, Examples, and Elementary Properties. Homomorphisms, Isomorphisms, and Cyclic Groups. Cosets and Lagrange's Theorem. The RSA Cipher (Optional). Elements of Coding Theory. The Hamming Metric. The Parity-Check and Generator Matrices. Group Codes: Decoding with Coset Leaders. Hamming Matrices. Counting and Equivalence: Burnside's Theorem. The Cycle Index. The Pattern Inventory: Polya's Method of Enumeration. Summary and Historical Review. 17. Finite Fields and Combinatorial Designs. Polynomial Rings. Irreducible Polynomials: Finite Fields. Latin Squares. Finite Geometries and Affine Planes. Block Designs and Projective Planes. Summary and Historical Review. Appendices. Exponential and Logarithmic Functions. Matrices, Matrix Operations, and Determinants. Countable and Uncountable Sets. Solutions. Index.

A characterization of rational singularities in terms of injectivity of Frobenius maps

Abstract: The notions of F -rational and F -regular rings are defined via tight closure, which is a closure operation for ideals in a commutative ring of positive characteristic. The geometric significance of these notions has persisted, and K. E. Smith proved that F -rational rings have rational singularities. We now ask about the converse implication. The answer to this question is yes and no. For a fixed positive characteristic, there is a rational singularity which is not F -rational, so the answer is no. In this paper, however, we aim to show that the answer is yes in the following sense: If a ring of characteristic zero has rational singularity, then its modulo p reduction is F -rational for almost all characteristic p . This result leads us to the correspondence of F -regular rings and log terminal singularities.

Separative cancellation for projective modules over exchange rings

Abstract: A separative ring is one whose finitely generated projective modules satisfy the propertyA⊕A⋟A⊕B⋟B⊕B⇒A⋟B. This condition is shown to provide a key to a number of outstanding cancellation problems for finitely generated projective modules over exchange rings. It is shown that the class of separative exchange rings is very broad, and, notably, closed under extensions of ideals by factor rings. That is, if an exchange ringR has an idealI withI andR/I both separative, thenR is separative.

Cellular Resolutions of Monomial Modules

Abstract: We construct a canonical free resolution for arbitrary monomial modules and lattice ideals. This includes monomial ideals and defining ideals of toric varieties, and it generalizes our joint results with Irena Peeva for generic ideals [BPS],[PS]. Introduction Given a field k, we consider the Laurent polynomial ring T = k[x±1 1 , . . . , x ±1 n ] as a module over the polynomial ring S = k[x1, . . . , xn]. The module structure comes from the natural inclusion of semigroup algebras S = k[N] ⊂ k[Z] = T . A monomial module is an S-submodule of T which is generated by monomials x = x1 1 · · ·xan n , a ∈ Z. Of special interest are the two cases when M has a minimal monomial generating set which is either finite or forms a group under multiplication. In the first case M is isomorphic to a monomial ideal in S. In the second case M coincides with the lattice module ML := S {x | a ∈ L} = k {x | b ∈ N + L} ⊂ T. for some sublattice L ⊂ Z whose intersection with N is the origin 0 = (0, . . . , 0). We shall derive free resolutions of M from regular cell complexes whose vertices are the generators of M and whose faces are labeled by the least common multiples of their vertices. The basic theory of such cellular resolutions is developed in Section 1. Our main result is the construction of the hull resolution in Section 2. We rescale the exponents of the monomials in M , so that their convex hull in R is a polyhedron Pt whose bounded faces support a free resolution of M . This resolution is new and interesting even for monomial ideals. It need not be minimal, but, unlike minimal resolutions, it respects symmetry and is free from arbitrary choices. In Section 3 we relate the lattice module ML to the Z/L-graded lattice ideal IL = 〈 x − x | a− b ∈ L 〉 ⊂ S. This class of ideals includes ideals defining toric varieties. We express the cyclic Smodule S/IL as the quotient of the infinitely generated S-module ML by the action of L. In fact, we like to think of ML as the “universal cover” of IL. Many questions about IL can thus be reduced to questions about ML. In particular, we obtain the hull resolution of a lattice ideal IL by taking the hull resolution of ML modulo L. This paper is inspired by the work of Barany, Howe and Scarf [BHS] who introduced the polyhedron Pt in the context of integer programming. The hull resolution generalizes results in [BPS] for generic monomial ideals and in [PS] for generic lattice ideals. In these generic cases the hull resolution is minimal.

Ring routing and wavelength translation

Abstract: Let G be the digraph consisting of two oppositely-directed rings on the same set of n nodes. We provide a polynomialtime algorithm which, given a list of demands-each requiring a path from a specified source node to a specified target node-routes the demands so as to minimize the largest number of paths through any of the 2n directed links of G. The algorithm makes use of a partial linear relaxation and rounding technique which together, somewhat surprisingly, produce an exact solution. The problem arises in an optical communications network with wavelength division multiplexing (WDM), configured as a ring. Such a network features a fixed number of wavelengths, each of which (at the optical level) can sup port a single path of high bandwidth through a given link. If there is no “wavelength translation” available, so that each demand is restricted to a single wavelength, then the combined routing and wavelength assignment problem is NPcomplete. Our results imply, however, that the presence of even a single wavelength translator (at any node) guarantees both full capacity and polynomial-time optimizabihty. Single-translator sufficiency in the ring is a special case of a simple criterion which, given a set of nodes in an arbitrary WDM network, determines whether wavelength translators on those nodes allow the network to run at maximum capacity. Although the problem of minimizing the cardinahty of this set is NP-complete (even in the planar case), the high cost of wavelength translators can be expected to make the criterion a useful tool.

Witt Rings in Real Algebraic Geometry

Abstract: In the first section we define the Witt ring W(A) of a commutative ring A and compare it with the group K 0(A). In particular, if V is an algebraic set over a real closed field R, we show that the Witt ring W(S 0(V)) coincides with K 0(S 0(V)) (≃ KO(V) if R = ℝ). The second section is devoted to the result of Mahe concerning the separation of the semi-algebraically connected components of V by the signatures of elements of W(P(V)). In the third section we prove that the morphism W(P(V))[1/2] → W(S 0(V))[1/2], induced by the inclusion P(V) → S 0(V), is surjective. This is part of the result of Brumfiel, which asserts that this morphism is actually an isomorphism.

Cohen–Macaulay Rings: Preface to the revised edition

Abstract: The main change in the revised edition is the new Chapter 10 on tight closure. This theory was created by Mel Hochster and Craig Huneke about ten years ago and is still strongly expanding. We treat the basic ideas, F -regular rings, and F -rational rings, including Smith's theorem by which F -rationality implies pseudo-rationality. Among the numerous applications of tight closure we have selected the Briancon–Skoda theorem and the theorem of Hochster and Huneke saying that equicharacteristic direct summands of regular rings are Cohen–Macaulay. To cover these applications, Section 8.4, which develops the technique of reduction to characteristic p , had to be rewritten. The title of Part III, no longer appropriate, has been changed. Another noteworthy addition are the theorems of Gotzmann in the new Section 4.3. We believe that Chapter 4 now treats all the basic theorems on Hilbert functions. Moreover, this chapter has been slightly reorganized. The new Section 5.5 contains a proof of Hochster's formula for the Betti numbers of a Stanley–Reisner ring since the free resolutions of such rings have recently received much attention. In the first edition the formula was used without proof. We are grateful to all the readers of the first edition who have suggested corrections and improvements. Our special thanks go to L. Avramov, A. Conca, S. Iyengar, R. Y. Sharp, B. Ulrich, and K.-i. Watanabe.

Ring network for sharing protection resource by working communication paths

Abstract: In a ring topology network, a number of nodes interconnect transmission links to form first and second working rings and first and second optical protection rings in a ring topology. Multiple working paths are established on each working ring and multiple protection paths are established on each protection ring corresponding to the working paths. A first working path spans across first and second nodes for transmission of a signal in a first direction of the ring topology, and a second working path of the second working ring spans across the first and second nodes for transmission of a signal in a second direction of the ring topology opposite to the first direction. A first protection path on the first protection ring spans across the first and second nodes for transmission of a signal in the second direction of the ring topology, and a second protection path of the second protection ring spans across the first and second nodes for transmission of a signal in the first direction of the ring topology. The first and second nodes normally use the first and second working paths, respectively. Responsive to a failure of one of the first and second working paths, the first and second nodes use a corresponding one of the first and second protection paths, instead of the failed working path.

Rings with Auslander Dualizing Complexes

Abstract: A ring with an Auslander dualizing complex is a generalization of an Auslander-Gorenstein ring. We show that many results which hold for Auslander-Gorenstein rings also hold in the more general setting. On the other hand we give criteria for existence of Auslander dualizing complexes which show these occur quite frequently. The most powerful tool we use is the Local Duality Theorem for connected graded algebras over a field. Filtrations allow the transfer of results to non-graded algebras. We also prove some results of a categorical nature, most notably the functoriality of rigid dualizing complexes.

Linear estimate for the number of zeros of abelian integrals with cubic hamiltonians

Abstract: An explicit upper bound is derived for the number of the zeros of the integral of degree n polynomials f, g, on the open interval for which the cubic curve contains an oval. The proof exploits the properties of the Picard-Fuchs system satisfied by the four basic integrals , i,j=0,1, generating the module of complete Abelian integrals I(h) (over the ring of polynomials in h).

A Planar Case of the n + 1 Body Problem: the 'Ring’ Problem

Abstract: Our intention in this article is to present a new model for the investigation of the motion of a particle of negligible mass in a multibody surrounding The proposed general planar configuration consists of ν = n-1 primaries arranged in equal arcs on an ideal ring and a central body of different mass located at the centre of mass of the system We formulate the general equations of motion and we study the stationary solutions and the zero-velocity contours for various values of ν

Equivariant de Rham Theory and Graphs

Abstract: Goresky, Kottwitz and MacPherson have recently shown that the computation of the equivariant cohomology ring of a G-manifold can be reduced to a computation in graph theory. This opens up the possibility that many of the fundamental theorems in equivariant de Rham theory may, on closer inspection, turn out simply to be theorems about graphs. In this paper we show that for some familiar theorems, this is indeed the case.

Resonant Hypergeometric Systems and Mirror Symmetry

Abstract: In Part I the Γ-series of [11] are adapted so that they give solutions for certain resonant systems of Gel’fand-Kapranov-Zelevinsky hypergeometric differential equations. For this some complex parameters in the Γseries are replaced by nilpotent elements from a ring RA,T . The adapted Γ-series is a function ΨT,β with values in the finite dimensional vector space RA,T ⊗ZC . Part II describes applications of these results in the context of toric Mirror Symmetry. Building on Batyrev’s work [2] we show that a certain relative cohomology module H(T rel Zs−1) is a GKZ hypergeometric D-module which over an appropriate domain is isomorphic to the trivial D-module RA,T ⊗ OT, where OT is the sheaf of holomorphic functions on this domain. The isomorphism is explicitly given by adapted Γ-series. As a result one finds the periods of a holomorphic differential form of degree d on a d-dimensional Calabi-Yau manifold, which are needed for the B-model side input to Mirror Symmetry. Relating our work with that of Batyrev and Borisov [3] we interpret the ring RA,T as the cohomology ring of a toric variety and a certain principal ideal in it as a subring of the Chow ring of a Calabi-Yau complete intersection. This interpretation takes place on the A-model side of Mirror Symmetry.

Countable linear transformations are clean

Abstract: It is shown that every linear transformation on a vector space of countable dimension is the sum of a unit and an idempotent. An element in a ring R is called clean in R if it is the sum of a unit and an idempotent, and the ring itself is called clean if every element is clean. Every clean ring is an exchange ring and, if R has central idempotents, R is an exchange ring if and only if it is clean [2, Proposition 1.8]. Camillo and Yu [1, Theorem 9] have shown that a ring is semiperfect if and only if it is clean and has no infinite orthogonal family of idempotents. Our main result is the following theorem which answers a question of P. Ara. Theorem. If VD is a vector space of countably infinite dimension over a division ring D, then end(VD) is clean. A ring R is called unit regular if, for each a E R, there exists a unit u E R such that aua = a. Camillo and Yu [1, Theorem 5] show that every unit regular ring is clean. The Theorem shows that the converse is not true. Corollary. There exists a (von Neumann) regular, right self-injective, clean ring which is not unit regular. Proof. The ring end(VD) in the Theorem suffices because it is not unit regular. In fact, it is not even Dedekind finite (ab= 1 implies ba= 1). D The proof of the Theorem employs several preliminary lemmas. Throughout this paper D always denotes a division ring and VD is always a vector space of countably infinite dimension over D. If {xI, x2, ... } is a basis of VD, the linear transformation a: V -* V given by a(xi) = xi+, for each i is called a shift operator on V. Lemma 1. Every shift operator on VD is clean in end(VD). Received by the editors July 16, 1996. 1991 Mathematics Subject Classification. Primary 16S50; Secondary 16E50, 16U99.

Algebraic Coset Conformal field theories

Abstract: All unitary Rational Conformal Field Theories (RCFT) are conjectured to be related to unitary coset Conformal Field Theories, i.e., gauged Wess-Zumino-Witten (WZW) models with compact gauge groups. In this paper we use subfactor theory and ideas of algebraic quantum field theory to approach coset Conformal Field Theories. Two conjectures are formulated and their consequences are discussed. Some results are presented which prove the conjectures in special cases. In particular, one of the results states that a class of representations of coset $W_N$ ($N\geq 3$) algebras with critical parameters are irreducible, and under the natural compositions (Connes' fusion), they generate a finite dimensional fusion ring whose structure constants are completely determined, thus proving a long-standing conjecture about the representations of these algebras.

Gorenstein injective and projective complexes

Abstract: In this article we extend the notion of Gorenstein injective and projective modules to that of complexes and characterize such complexes. We prove that over an n-Gorenstein ring every complex has a Gorenstein injective envelope and we show that every such envelope is a quasi-isomorphim.. When the ring is commutative, local and Gorenstein, Auslander announced that every finitely generated R-module has a finitely generated Gorenstein projective cover. We show that every bounded above complex having all terms finitely generated over such a ring has a Gorenstein projective cover and we show that these covers are quasi-isomorphisms.

An algebraic approach to Wigner's unitary-antiunitary theorem

Abstract: We present an operator algebraic approach to Wigner's unitary-antiunitary theorem using some classical results from ring theory. To show how effective this approach is, we prove a generalization of this celebrated theorem for Hilbert modules over matrix algebras. We also present a Wigner-type result for maps on prime C *-algebras.

Mixed Bruhat operators and Yang-Baxter equations for Weyl groups

Abstract: We introduce and study a family of operators which act in the span of a Weyl group $W$ and provide a multi-parameter solution to the quantum Yang-Baxter equations of the corresponding type. Our operators generalize the "quantum Bruhat operators" that appear in the explicit description of the multiplicative structure of the (small) quantum cohomology ring of $G/B$. The main combinatorial applications concern the "tilted Bruhat order," a graded poset whose unique minimal element is an arbitrarily chosen element $w\in W$. (The ordinary Bruhat order corresponds to the case $w=1$.) Using the mixed Bruhat operators, we prove that these posets are lexicographically shellable, and every interval in a tilted Bruhat order is Eulerian. This generalizes well known results of Verma, Bjorner, Wachs, and Dyer.

On Cohen-Macaulay and Gorenstein simplicial affine semigroups

Abstract: We give arithmetic characterizations which allow us to determine algorithmically when the semigroup ring associated to a simplicial affine semigroup is Cohen-Macaulay and/or Gorenstein. These characterizations are then used to provide information about presentations of this kind of semigroup and, in particular, to obtain bounds for the cardinality of their minimal presentations. Finally, we show that these bounds are reached for semigroups with maximal codimension.

Unramified cohomology of quadrics, I

Abstract: Given a quadric X over a field F of characteristic ≠ 2, we compute the kernel and cokernel of the natural map in degree 4 from the mod 2 Galois cohomology of F to the unramified mod 2 cohomology of F ( X ), when dim X 10 and in several smaller-dimensional cases. Applications of these results to real quadrics and to the unramified Witt ring are given.