Showing papers on "Ring (mathematics) published in 2006"
Book•
01 Jan 2006
TL;DR: Approximations and endomorphism algebras of modules have been studied extensively in the literature since 2006 as mentioned in this paper, with a focus on the impossibility of classification for modules over general rings.
Abstract: This second, revised and substantially extended edition of Approximations and Endomorphism Algebras of Modules reflects both the depth and the width of recent developments in the area since the first edition appeared in 2006. The new division of the monograph into two volumes roughly corresponds to its two central topics, approximation theory (Volume 1) and realization theorems for modules (Volume 2). It is a widely accepted fact that the category of all modules over a general associative ring is too complex to admit classification. Unless the ring is of finite representation type we must limit attempts at classification to some restricted subcategories of modules. The wild character of the category of all modules, or of one of its subcategories C, is often indicated by the presence of a realization theorem, that is, by the fact that any reasonable algebra is isomorphic to the endomorphism algebra of a module from C. This results in the existence of pathological direct sum decompositions, and these are generally viewed as obstacles to classification. In order to overcome this problem, the approximation theory of modules has been developed. The idea here is to select suitable subcategories C whose modules can be classified, and then to approximate arbitrary modules by those from C. These approximations are neither unique nor functorial in general, but there is a rich supply available appropriate to the requirements of various particular applications. The authors bring the two theories together. The first volume, Approximations, sets the scene in Part I by introducing the main classes of modules relevant here: the S-complete, pure-injective, Mittag-Leffler, and slender modules. Parts II and III of the first volume develop the key methods of approximation theory. Some of the recent applications to the structure of modules are also presented here, notably for tilting, cotilting, Baer, and Mittag-Leffler modules. In the second volume, Predictions, further basic instruments are introduced: the prediction principles, and their applications to proving realization theorems. Moreover, tools are developed there for answering problems motivated in algebraic topology. The authors concentrate on the impossibility of classification for modules over general rings. The wild character of many categories C of modules is documented here by the realization theorems that represent critical R-algebras over commutative rings R as endomorphism algebras of modules from C. The monograph starts from basic facts and gradually develops the theory towards its present frontiers. It is suitable both for graduate students interested in algebra and for experts in module and representation theory.
489 citations
04 Mar 2006
TL;DR: In this article, the authors propose new definitions of anonymity and unforgeability for ring signatures, and show two constructions of ring signature schemes in the standard model: one based on generic assumptions which satisfies their strongest definitions of security, and a second, more efficient scheme achieving weaker security guarantees and more limited functionality.
Abstract: Ring signatures, first introduced by Rivest, Shamir, and Tauman, enable a user to sign a message so that a ring of possible signers (of which the user is a member) is identified, without revealing exactly which member of that ring actually generated the signature. In contrast to group signatures, ring signatures are completely “ad-hoc” and do not require any central authority or coordination among the various users (indeed, users do not even need to be aware of each other); furthermore, ring signature schemes grant users fine-grained control over the level of anonymity associated with any particular signature.
This paper has two main areas of focus. First, we examine previous definitions of security for ring signature schemes and suggest that most of these prior definitions are too weak, in the sense that they do not take into account certain realistic attacks. We propose new definitions of anonymity and unforgeability which address these threats, and then give separation results proving that our new notions are strictly stronger than previous ones. Next, we show two constructions of ring signature schemes in the standard model: one based on generic assumptions which satisfies our strongest definitions of security, and a second, more efficient scheme achieving weaker security guarantees and more limited functionality. These are the first constructions of ring signature schemes that do not rely on random oracles or ideal ciphers.
317 citations
TL;DR: In this paper, a representation-theoretic construction of the cluster algebra structure on the ring ℂ[N] of polynomial functions on a maximal unipotent subgroup N of a complex Lie group of type Δ is presented.
Abstract: Let Λ be a preprojective algebra of simply laced Dynkin type Δ. We study maximal rigid Λ-modules, their endomorphism algebras and a mutation operation on these modules. This leads to a representation-theoretic construction of the cluster algebra structure on the ring ℂ[N] of polynomial functions on a maximal unipotent subgroup N of a complex Lie group of type Δ. As an application we obtain that all cluster monomials of ℂ[N] belong to the dual semicanonical basis.
270 citations
TL;DR: The SL(2, ℤ)-representation π on the center of the restricted quantum group at the primitive 2pth root of unity is shown to be equivalent to the SL( 2, ↦)-representations on the extended characters of the logarithmic (1, p) conformal field theory model in this article.
Abstract: The SL(2, ℤ)-representation π on the center of the restricted quantum group at the primitive 2pth root of unity is shown to be equivalent to the SL(2, ℤ)-representation on the extended characters of the logarithmic (1, p) conformal field theory model. The multiplicative Jordan decomposition of the ribbon element determines the decomposition of π into a ``pointwise'' product of two commuting SL(2, ℤ)-representations, one of which restricts to the Grothendieck ring; this restriction is equivalent to the SL(2, ℤ)-representation on the (1, p)-characters, related to the fusion algebra via a nonsemisimple Verlinde formula. The Grothendieck ring of at the primitive 2pth root of unity is shown to coincide with the fusion algebra of the (1, p) logarithmic conformal field theory model. As a by-product, we derive q-binomial identities implied by the fusion algebra realized in the center of .
268 citations
Posted Content•
TL;DR: In this paper, a general regular black ring solution with two angular momenta is presented, found by the inverse scattering problem method, where the mass, angular momentsa and the event horizon volume are given explicitly as functions of the metric parameters.
Abstract: General regular black ring solution with two angular momenta is presented, found by the inverse scattering problem method The mass, angular momenta and the event horizon volume are given explicitly as functions of the metric parameters
256 citations
TL;DR: A black ring is a five-dimensional black hole with an event horizon of topology S 1 x S 2 as discussed by the authors, and it is defined in general relativity and string theory.
Abstract: A black ring is a five-dimensional black hole with an event horizon of topology S1 x S2 We provide an introduction to the description of black rings in general relativity and string theory Novel aspects of the presentation include a new approach to constructing black ring coordinates and a critical review of black ring microscopics
189 citations
Posted Content•
TL;DR: In this paper, the authors introduce a new general construction, called the amalgamated duplication of a ring $R$ along an ideal module $E$ that they assume to be an ideal in some overring of $R$.
Abstract: We introduce a new general construction, denoted by $R\JoinE$, called the amalgamated duplication of a ring $R$ along an $R$--module $E$, that we assume to be an ideal in some overring of $R$. (Note that, when $E^2 =0$, $R\JoinE$ coincides with the Nagata's idealization $R\ltimes E$.)
After discussing the main properties of the amalgamated duplication $R\JoinE$ in relation with pullback--type constructions, we restrict our investigation to the study of $R\JoinE$ when $E$ is an ideal of $R$.
Special attention is devoted to the ideal-theoretic properties of $R\JoinE$ and to the topological structure of its prime spectrum.
144 citations
TL;DR: In this article, a new theory for the formation of rR1 ring structures was proposed, i.e. for ring structures with both an inner and an outer ring, the latter having the form of "8".
Abstract: We propose a new theory for the formation of rR1 ring structures, i.e. for ring structures with both an inner and an outer ring, the latter having the form of "8". We propose that these rings are formed by material from the stable and unstable invariant manifolds associated with the Lyapunov orbits around the equilibrium points of a barred galaxy. We discuss the shape and velocity structure of the rings thus formed and argue that they are in agreement with the observed properties of rR1 structures.
130 citations
TL;DR: In this article, it was shown that all reversible rings are McCoy, generalizing the fact that both commutative and reduced rings are also McCoy, and that semi-commutative rings do have a property close to the McCoy condition.
122 citations
TL;DR: In this paper, the authors proposed a new theory for the formation of rR_1 ring structures, i.e. for ring structures with both an inner and an outer ring, the latter having the form of ''8''.
Abstract: We propose a new theory for the formation of rR_1 ring structures, i.e. for ring structures with both an inner and an outer ring, the latter having the form of ``8''. We propose that these rings are formed by material from the stable and unstable invariant manifolds associated with the Lyapunov orbits around the equilibrium points of a barred galaxy. We discuss the shape and velocity structure of the rings thus formed and argue that they are in agreement with the observed properties of rR_1 structures.
118 citations
TL;DR: This work formalizes the notion of a ring signature, which makes it possible to specify a set of possible signers without revealing which member actually produced the signature, and presents the presentation of efficient constructions of ring signatures.
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TL;DR: The F-thresholds are characteristic p analogs of the jumping coefficients for multiplier ideals in characteristic zero as discussed by the authors, and they have been shown to be rational and discrete in a regular and F-finite ring.
Abstract: The F-thresholds are characteristic p analogs of the jumping coefficients for multiplier ideals in characteristic zero. In this article we give an alternative description of the F-thresholds of an ideal in a regular and F--finite ring $R$. This enables us to settle two open questions posed in [Mustata, Takagi, Watanabe: F-thresholds and Bernstein-Sato polynomials], namely we show that the F-thresholds are rational and discrete.
TL;DR: In this paper, it was shown that a 2-primal ring R is weakly pm if and only if Max (R ) is a retract of SSpec (R ), where SSpec is the space of strongly prime ideals of R and R is a symmetric ring.
TL;DR: Weak Armendariz rings as mentioned in this paper are a generalization of semicommutative rings and are shown to be weak armendariz if and only if for any n, the n-by-n upper triangular matrix ring T n (R) is weak ARM.
Abstract: We introduce weak Armendariz rings which are a generalization of semicommutative rings and Armendariz rings, and investigate their properties. Moreover, we prove that a ring R is weak Armendariz if and only if for any n, the n-by-n upper triangular matrix ring T n (R) is weak Armendariz. If R is semicommutative, then it is proven that the polynomial ring R[x] over R and the ring R[x]/(x n ), where (x n ) is the ideal generated by x n and n is a positive integer, are weak Armendariz.
TL;DR: In this article, an automatic approach consisting of a spatial filter, a classification module and an estimation module is proposed to validate both real and simulated data, and three types of typical defect patterns: linear scratch, circular ring, and an elliptical zone can be successfully extracted and classified.
Abstract: The detection of process problems and parameter drift at an early stage is crucial to successful semiconductor manufacture. The defect patterns on the wafer can act as an important source of information for quality engineers allowing them to isolate production problems. Traditionally, defect recognition is performed by quality engineers using a scanning electron microscope. This manual approach is not only expensive and time consuming but also it leads to high misidentification levels. In this paper, an automatic approach consisting of a spatial filter, a classification module and an estimation module is proposed to validate both real and simulated data. Experimental results show that three types of typical defect patterns: (i) a linear scratch; (ii) a circular ring; and (iii) an elliptical zone can be successfully extracted and classified. A Gaussian EM algorithm is used to estimate the elliptic and linear patterns, and a spherical-shell algorithm is used to estimate ring patterns. Furthermore, both conv...
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TL;DR: The notion of projective modules over non-noetherian commutative rings was introduced and investigated in this article, where a semidualizing projective module is considered.
Abstract: We introduce and investigate the notion of $\gc$-projective modules over (possibly non-noetherian) commutative rings, where $C$ is a semidualizing module. This extends Holm and J{\o}rgensen's notion of $C$-Gorenstein projective modules to the non-noetherian setting and generalizes projective and Gorenstein projective modules within this setting. We then study the resulting modules of finite $\gc$-projective dimension, showing in particular that they admit $\gc$-projective approximations, a generalization of the maximal Cohen-Macaulay approximations of Auslander and Buchweitz. Over a local (noetherian) ring, we provide necessary and sufficient conditions for a $G_C$-approximation to be minimal.
01 Jan 2006
TL;DR: The concept of motivic integration was introduced by Kontsevich and Batyrev as mentioned in this paper, who constructed a measure on the arc space of an algebraic variety, the motivic measure, with the subtle and crucial property that it takes values not in $\mathbb{R}$, but in the Grothendieck ring of algebraic varieties.
Abstract: The concept of motivic integration was invented by Kontsevich to show that birationally equivalent Calabi-Yau manifolds have the same Hodge numbers. He constructed a certain measure on the arc space of an algebraic variety, the motivic measure, with the subtle and crucial property that it takes values not in $\mathbb{R}$, but in the Grothendieck ring of algebraic varieties. A whole theory on this subject was then developed by Denef and Loeser in various papers, with several applications. Batyrev introduced with motivic integration techniques new singularity invariants, the stringy invariants, for algebraic varieties with mild singularities, more precisely log terminal singularities. He used them for instance to formulate a topological Mirror Symmetry test for pairs of singular Calabi-Yau varieties. We generalized these invariants to almost arbitrary singular varieties, assuming Mori's Minimal Model Program. The aim of these notes is to provide a gentle introduction to these concepts. There exist already good surveys by Denef-Loeser [DL8] and Looijenga [Loo], and a nice elementary introduction by Craw [Cr]. Here we merely want to explain the basic concepts and first results, including the $p$-adic number theoretic pre-history of the theory, and to provide concrete examples. The text is a slightly adapted version of the 'extended abstract' of the author's talks at the 12th MSJ-IRI "Singularity Theory and Its Applications" (2003) in Sapporo. At the end we included a list of various recent results.
TL;DR: In this paper, a generalized approach for determination of transmittance (transfer function) in Z-domain of optical waveguide based ring resonator is introduced, and the simulated results of single and double ring architectures are compared with those of previously published results.
TL;DR: In this article, a general theory of ''Specht modules'' for Hecke algebras of finite type was obtained for a finite Weyl group with invertible primes.
Abstract: Let $\cH$ be the one-parameter Hecke algebra associated to a finite Weyl group $W$, defined over a ground ring in which ``bad'' primes for $W$ are invertible. Using deep properties of the Kazhdan--Lusztig basis of $\cH$ and Lusztig's $\ba$-function, we show that $\cH$ has a natural cellular structure in the sense of Graham and Lehrer. Thus, we obtain a general theory of ``Specht modules'' for Hecke algebras of finite type. Previously, a general cellular structure was only known to exist in types $A_n$ and $B_n$.
TL;DR: In this paper, it was shown that repeated-root cyclic codes over a finite chain ring are in general not principally generated, and that negacyclic codes are not necessarily generated.
TL;DR: In this article, the authors define a ring R to be weakly clean if each element of R can be written as either the sum or difference of a unit and an idempotent.
Abstract: Let R be a commutative ring with identity. Nicholson defined R to be clean if each element of R is the sum of a unit and an idempotent. In this paper we study two related classes of rings. We define a ring R to be weakly clean if each element of R can be written as either the sum or difference of a unit and an idempotent and following McGovern we say that R is almost clean if each element of R is the sum of a nonzero-divisor and an idempotent.
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TL;DR: In this article, the notion of lax coring was introduced, which generalizes Wisbauer's notion of weak coring, and several duality results were given, and Galois theory for partial Hopf algebras was developed.
Abstract: We introduce partial (co)actions of a Hopf algebra $H$ on an algebra. To this end, we introduce first the notion of lax coring, generalizing Wisbauer's notion of weak coring. We also have the dual notion of lax ring. Several duality results are given, and we develop Galois theory for partial $H$-comodule algebras.
Journal Article•
TL;DR: Each ideal has a unique distinguished set of generators that characterizes any cyclic code and is associated with each ideal in the ring Z4[x]/(xn-1).
Abstract: Results are presented on the generators of ideals in the ring Z4[x]/(xn-1). In particular, each ideal (cyclic code) has a unique distinguished set of generators that characterizes any cyclic code. ...
TL;DR: The problem of classifying the objects in a commutative local uniserial ring with radical factor field k up to isomorphism has been studied in this article, where it has been shown that S ( Λ ) is controlled k-wild with a single control object I ∈ S( Λ ).
TL;DR: In this paper, the influence of specimen geometry on the results of the restrained ring test considering three conditions: (1) uniform shrinkage of concrete ring, (2) shrinkage caused by drying from the top and bottom surfaces of the concrete ring and (3) shrinking caused by dripping from the outer circumference of the concave ring.
Abstract: Over the last decade, the restrained ring test has trequently been used to assess the cracking susceptibility of a concrete mixture when it is restrained from shrinking treely. Despite the trequent use of the ring test, limited analysis has been performed to understand how the specimen geometry influences the results of the test. This paper discusses the influence of specimen geometry on the results of the ring test considering three conditions: (1) uniform shrinkage of the concrete ring, (2) shrinkage caused by drying from the top and bottom surfaces of the concrete ring, and (3) shrinkage caused by drying from the outer circumference of the concrete ring. The role of moisture gradients, thickness of the concrete and the restraining (i.e., steel) rings, and the stiffness of concrete are considered in a series of numerical simulations. Results from these simulations can enable better selection of test specimen geometries and interpretation of the results from the ring test. Analytical expressions are provided to use for determining the geometry of the ring specimen that better simulates specific field conditions while providing the most useful information from the test.
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TL;DR: A survey of the known properties of Iwasawa algebras is given in this article, where a number of open questions are also stated, as well as a discussion of the properties of complete group rings of compact p-adic analytic groups with coefficients.
Abstract: This is a survey of the known properties of Iwasawa algebras, i.e., completed group rings of compact p-adic analytic groups with coefficients the ringZp of p-adic integers or the field Fp of p elements. A number of open questions are also stated.
TL;DR: The logarithmic cohomology operation as discussed by the authors is a homotopy-theoretic analogue of the above, where R is a commutative S-algebra and Bousfield localization with respect to a Morava K-theory.
Abstract: Recall that if R is a commutative ring, then the set R× ⊂ R of invertible elements of R is naturally an abelian group under multiplication. This construction is a functor from commutative rings to abelian groups. In general, there is no obvious relation between the additive group of a ring R and the multiplicative group of units R×. However, under certain circumstances one can define a homomorphism from a subgroup of R× to a suitable completion of R, e.g., the natural logarithm Q>0 → R, or the p-adic logarithm (1 + pZp) → Zp. The “logarithmic cohomology operation” is a homotopy-theoretic analogue of the above, where R is a commutative S-algebra and “completion” is Bousfield localization with respect to a Morava K-theory. The purpose of this paper is to give a formula for the logarithmic operation (in certain contexts) in terms of power operations. Before giving our results we briefly explain some of the concepts involved.
TL;DR: In this article, the authors present a scheme for creating macroscopic superpositions of the direction of superfluid flow around a loop using the Bose-Einstein condensates.
Abstract: We present a scheme for creating macroscopic superpositions of the direction of superfluid flow around a loop. Using the Bose–Hubbard model, we study an array of Bose–Einstein condensates (BECs) trapped in optical potentials and coupled to one another to form a ring. By rotating the ring so that each particle acquires on average half a quantum of superfluid flow, it is possible to create a multiparticle superposition of all the particles rotating and all the particles stationary. Under certain conditions it is possible to scale up the number of particles to form a macroscopic superposition. The simplicity of the model has allowed us to study macroscopic superpositions at an atomic level for different variables. Here, we concentrate on the tunnelling strength between the potentials. Further investigation remains important, because it could lead us to making an ultra-precise quantum-limited gyroscope.
TL;DR: The power structure over the Grothendieck (semi)ring of complex quasi-projective varieties constructed by the authors as mentioned in this paper is used to express the generating series of classes of Hilbert schemes of zero-dimensional subschemes on a smooth quasi projective variety as an exponent of that for the complex affine space of the same dimension.
Abstract: The power structure over the Grothendieck (semi)ring of complex quasi-projective varieties constructed by the authors is used to express the generating series of classes of Hilbert schemes of zero-dimensional subschemes on a smooth quasi-projective variety as an exponent of that for the complex affine space of the same dimension. Specializations of this relation give formulae for generating series of such invariants of the Hilbert schemes of points as the Euler characteristic and the Hodge-Deligne polynomial.
TL;DR: In this paper, it was shown that a lifting module is lifted if and only if every generalized supplement submodule is a GAS-module and satisfies DCC on small submodules.
Abstract: Let $R$ be a ring and $M$ a right $R$-module. It is shown that: (1) $M$ is Artinian if and only if $M$ is a GAS-module and satisfies DCC on generalized supplement submodules and on small submodules; (2) if $M$ satisfies ACC on small submodules, then $M$ is a lifting module if and only if $M$ is a GAS-module and every generalized supplement submodule is a direct summand of $M$ if and only if $M$ satisfies $(P^{*})$; (3) $R$ is semilocal if and only if every cyclic module is a GWS-module.