Showing papers on "Ring (mathematics) published in 1992"
TL;DR: A new probabilistic algorithm for factoring univariate polynomials over finite fields is presented, with main technical innovation a new way to compute Frobenius and trace maps in the ring of polynmials modulo the polynomial to be factored.
Abstract: A new probabilistic algorithm for factoring univariate polynomials over finite fields is presented. To factor a polynomial of degreen overF
q
, the number of arithmetic operations inF
q
isO((n
2+nlogq). (logn)2 loglogn). The main technical innovation is a new way to compute Frobenius and trace maps in the ring of polynomials modulo the polynomial to be factored.
202 citations
TL;DR: It is shown that, over an arbitrary ring, the functions computed by polynomial-size algebraic formulas are also computed by algebraic straight-line programs that use only three registers, which is an improvement over previous methods that require the number of registers to be logarithmic in the size of the formulas.
Abstract: It is shown that, over an arbitrary ring, the functions computed by polynomial-size algebraic formulas are also computed by polynomial-length algebraic straight-line programs that use only three registers. This was previously known for Boolean formulas [D. A. Barrington, J. Comput. System Sci., 38 (1989), pp. 150–164], which are equivalent to algebraic formulas over the ring $GF(2)$. For formulas over arbitrary rings, the result is an improvement over previous methods that require the number of registers to be logarithmic in the size of the formulas in order to obtain polynomial-length straight-line programs. Moreover, the straight-line programs that arise in these constructions have the property that they consist of statements whose actions on the registers are linear and bijective. A consequence of this is that the problem of determining the iterated product of $n3 \times 3$ matrices is complete (under P-projections) for algebraic $NC^1 $. Also, when the ring is $GF(2)$, the programs that arise in the c...
186 citations
Journal Article•
TL;DR: In this paper, the authors classify simple flops on smooth three-folds, or equivalently, Gorenstein threefold singularities with irreducible small resolution, and apply invariant theory to Pinkham's construction of small resolutions.
Abstract: We classify simple flops on smooth threefolds, or equivalently, Gorenstein threefold singularities with irreducible small resolution. There are only six families of such singularities, distinguished by Koll{a}r's {\em length} invariant. The method is to apply invariant theory to Pinkham's construction of small resolutions. As a by-product, generators of the ring of invariants are given for the standard action of the Weyl group of each of the irreducible root systems.
167 citations
TL;DR: In this paper, the moduli space of classical minima for computing correlation functions involving twisted operators was described, and the complete ring of observables was computed for nonabelian orbifolds.
Abstract: We discuss the toplogical sigma model on an orbifold target space. We describe the moduli space of classical minima for computing correlation functions involving twisted operators, and show, through a detailed computation of an orbifold of ${\bf CP}^1$ by the dihedral group $D_{4},$ how to compute the complete ring of observables. Through this procedure, we compute all the rings from dihedral ${\bf CP}^1$ orbifolds; we note a similarity with rings derived from perturbed $D-$series superpotentials of the $A-D-E$ classification of $N = 2$ minimal models. We then consider ${\bf CP}^2/D_4,$ and show how the techniques of topological-anti-topological fusion might be used to compute twist field correlation functions for nonabelian orbifolds.
98 citations
23 Sep 1992
TL;DR: In this article, the topological Hochschild homology of a discrete ring is shown to agree with the MacLane homology for that ring, and it is shown that the topology of the discrete ring can be shown to be the same as that of the continuous ring.
Abstract: The topological Hochschild homology of a discrete ring is shown to agree with the MacLane homology of that ring.
94 citations
01 Jan 1992
TL;DR: In this article, the basic ring theoretic properties are described, and a number of questions and problems are raised, including quantum groups, quantum groups and quantized enveloping algebras.
Abstract: This article describes recent work on new classes of non-commutative algebras which have been dubbed quantum algebras, quantum groups, and quantized enveloping algebras. The basic ring theoretic properties are described, and a number of questions and problems are raised.
85 citations
TL;DR: The existence of these rings and modules leads to non-trivial constraints on the correlation functions and goes a long way toward solving two-dimensional quantum gravity models in the continuum approach.
69 citations
01 Mar 1992
TL;DR: In this article, it was shown that every additive centralizing mapping of a prime ring of characteristic not 2 is not commutative and therefore is not a commuting mapping, and this result holds for any additive mapping.
Abstract: Let R be a ring with center Z. A mapping f: R -* R is called centralizing (resp. commuting) if [f(x), x] E Z (resp. [f(x), x] = 0) for all x E R. In this paper we consider a more general case where a mapping f: R -E R satisfies [[f(x), x], x] = 0 for all x E R; it is shown that if R is a prime ring of characteristic not 2, then every additive mapping with this property is commuting. Let R be a ring with center Z. A mapping f of R into itself is called centralizing if [f(x), x] E Z for all x E R, where [u, v] denotes the commutator uv vu. In the special case where [f(x), x] = 0 for all x E R, the mapping f is said to be commuting. In [10] Posner initiated the study of centralizing mappings. He showed that if d is a centralizing derivation of a prime ring R, then either d = 0 or R is commutative. Over the last twenty years a lot of work has been done on centralizing and commuting mappings. We refer the reader to some recent papers [1, 2, 3, 4, 5, 7, 9, 1 1] where further references can be found. Recently Vukman [11] extended Posner's theorem by showing that if d is a derivation of a prime ring R of characteristic not 2, such that [[d(x), x], x] = 0 for all x E R, then d = 0 or R is commutative. In fact, in view of Posner's theorem he merely showed that d is commuting. It is our aim in this paper to prove that this conclusion holds for any additive mapping. More precisely, we prove the following result. Theorem 1. Let R be a prime ring of characteristic not 2. If an additive mapping f: R -R satisfies [[f(x), x], x] = 0 for all x E R, then [f(x), x] = 0 for all x E R (i.e., f is commuting). In particular, this theorem implies that every additive centralizing mapping of a prime ring of characteristic not 2 is actually commuting. However, in our forthcoming paper [4] it is shown that this result is true under some weaker hypothesis. In [4] we also proved the following result: If f is an additive commuting mapping of a prime ring R, then there exist an element A in C, the Received by the editors September 27, 1990. 1980 Mathematics Subject Classification (1985 Revision). Primary 16A 12, 1 6A70, 16A72.
64 citations
TL;DR: In this paper, a generalization of the Adler-Gel-fand-Dikii scheme is used to generate bi-Hamiltonian structures in two spatial dimensions, which can be used for the Kadomtsev-Petviashvili equation as well as other integrable equations in 2+1.
Abstract: A generalization of the Adler–Gel’fand–Dikii scheme is used to generate bi‐Hamiltonian structures in two spatial dimensions. In order to implement this scheme, a Hamiltonian theory is built over a noncommutative ring, namely the ring of formal pseudodifferential operators. Bi‐Hamiltonian structures generated in this way can be used for the Kadomtsev–Petviashvili equation as well as other integrable equations in 2+1.
63 citations
Book•
01 Jan 1992
TL;DR: Artinian rings with Morita duality (I) as mentioned in this paper, Azumaya's exact rings, and other types of rings with duality, including extensions and ring extensions.
Abstract: to Morita duality.- Morita duality and ring extensions.- Artinian rings with Morita duality (I).- Artinian rings (II) - Azumaya's exact rings.- Other types of rings with duality.
62 citations
Posted Content•
TL;DR: In this paper, an erratum that corrects an error in the proof of Proposition 4.3 in my paper "The Homogeneous Coordinate Ring of a Toric Variety" is presented.
Abstract: This submission consists of two papers: 1) an erratum that corrects an error in the proof of Proposition 4.3 in my paper "The Homogeneous Coordinate Ring of a Toric Variety", and 2) the original (unchanged) version of the paper, published in 1995. The original paper introduced the homogeneous coordinate ring of a toric variety (now called the total coordinate ring or Cox ring) and gave a quotient construction. The paper also studied sheaves on a toric variety, and in Section 4 described its automorphism group. The error in the proof of Proposition 4.3 resulted from the faulty assumption that a certain set of graded endomorphisms forms a ring; rather, it is a monoid under composition. The erratum notes this error and gives a correct proof of the proposition.
Proceedings Article•
01 Jan 1992
TL;DR: In this paper, the problem of decoding geometric Goppa codes can be reduced to solving the key congruence of a received word in an affine ring with complexity O(n 3 ), where n is the length of codewords.
Abstract: Decoding geometric Goppa codes can be reduced to solving the key congruence of a received word in an affine ring. If the codelength is smaller than the number of rational points on the curve, then this method can correct up to 1/2(d* − 1) − s errors, where d* is the designed minimum distance of the code and s is the Clifford defect. The affine ring with respect to a place P is the set of all rational functions which have no poles except at P, and it is somehow similar to a polynomial ring. For a special kind of geometric Goppa codes, namely C Ω (D, mP), the decoding algorithm is reduced to solving the key equation in the affine ring, which can be carried out by the subresultant sequence in the affine ring with complexity O(n 3 ), where n is the length of codewords
TL;DR: In this article, the authors studied Hopfian and co-Hopfian objects in categories like the category of rings, the module categories A-mod and mod-A for any ring A and showed that any Boolean ring A can be regarded as the ring A of clopen subsets of compact Hausdorff totally disconnected space X.
Abstract: The aim of the present paper is to study Hopfian and Co-Hopfian objects in categories like the category of rings, the module categories A-mod and mod-A for any ring A. Using Stone's representation theorem any Boolean ring can be regarded as the ring A of clopen subsets of compact Hausdorff totally disconnected space X. It turns out that the Boolean ring A will be Hopfian (resp. co-Hopfian) if and only if the space X is co-Hopfian (resp. Hopfian) in the category Top. For any compact Hausdorff space X let CR(X) (resp. CC(X)) denote the R (resp. C)-algebra of real (resp. complex) valued continuous functions on X. Using Gelfand's representation theorem we will prove that CR(X) (CC(X)) is Hopfian (respectively co-Hopfian) as an R(C) algebra if and only if X is co-Hopfian (respectively Hopfian) as an object of Top. We also study Hopfian and co-Hopfian compact topological manifolds.
TL;DR: In this article, the problem of decoding geometric Goppa codes was reduced to solving the key congruence of a received word in an affine ring, which can correct up to 1.2 (d*-L)/2 s errors, where l is the designed minimum distance of the code and s is the Clifford defect.
Abstract: Decoding geometric Goppa codes can be reduced to solving the key congruence of a received word in an affine ring. If the codelength is smaller than the number of rational points on the curve, then this method can correct up to 1.2 (d*-L)/2-s errors, where d* is the designed minimum distance of the code and s is the Clifford defect. The affine ring with respect to a place P is the set of all rational functions which have no poles except at P, and it is somehow similar to a polynomial ring. For a special kind of geometric Goppa code, namely C/sub Omega /(D,mP), the decoding algorithm is reduced to solving the key equation in the affine ring, which can be carried out by the subresultant sequence in the affine ring with complexity O(n/sup 3/), where n is the length of codewords. >
TL;DR: In this paper, the authors extend the classification of linear Jordan systems (triple systems and pairs) of hermitian type to quadratic Jordan systems over an arbitrary ring of scalars.
01 Jan 1992
TL;DR: In this paper, the authors construct a counter example to the conjecture of Johns that every right ideal is an annihilator in a right Noetherian ring in which every ideal is right Artinian.
Abstract: In this paper, we construct a counter example to a conjecture of Johns to the effect that a right Noetherian ring in which every right ideal is an annihilator is right Artinian. Our example requires the existence of a right Noetherian domain A (not a field) with a unique simple right module W such that WA is injective and A embeds in the endomorphism ring End( WA) . Then the counter example is the trivial extension R = A K W of A and W. The ring A exists by a theorem of Resco using a theorem of Cohn. Specifically, if D is any countable existentially closed field with center k, then the right and left principal ideal domain defined by A = D E?k k(x) , where k(x) is the field of rational functions, has the desired properties, with WA DA .
TL;DR: A quasi-projective right R -module M is a semi-artinian V-module if and only if for every m ∈ M and submodule N of mR, mR / N contains a nonzero M -injective submodule.
TL;DR: The structure of the complex cobordism ring of the flag variety of a compact connected Lie group is studied in this article, where an explicit procedure for determining products of basis elements is obtained.
Abstract: We study the structure of the complex cobordism ring of the flag variety of a compact connected Lie group. An explicit procedure for determining products of basis elements is obtained, generalizing the work of Bernstein-Gel'fand-Gel'fand on ordinary cohomology and of Kostant-Kumar on K-theory. Bott-Samelson resolutions are used to replace the classical basis of Schubert cells
TL;DR: Thea-invariants of graded algebras with straightening laws on upper semi-modular lattices and the Stanley-Reisner rings of shellable weighted simplicial complexes were derived in this article.
Abstract: Thea-invariant of a graded Cohen-Macaulay ring is the least degree of a generator of its graded canonical module. We compute thea-invariants of (i) graded algebras with straightening laws on upper semi-modular lattices and (ii) the Stanley-Reisner rings of shellable weighted simplicial complexes. The formulas obtained are applied to rings defined by determinantal and pfaffian ideals.
TL;DR: In this article, the ring of invariant polynomial functions on the general algebra of Cartan type is described explicitly and a criterion for orbits in to be closed is obtained, and it is proved that the commutator subgroup of the automorphism group in is stable.
Abstract: The ring of invariant polynomial functions on the general algebra of Cartan type is described explicitly. It is assumed that the ground field is algebraically closed and its characteristic is greater than 2. This result is used to prove that the variety of nilpotent elements in is an irreducible complete intersection and contains an open orbit whose complement consists of singular points. Moreover, a criterion for orbits in to be closed is obtained, and it is proved that the action of the commutator subgroup of the automorphism group in is stable.
01 Jan 1992
TL;DR: In this article, a dual formulation of representation theory of general quantum groups is given, and the rank of quantum groups in dual form and its connection with the partition function of simple quantum systems is explained.
Abstract: We give a dual formulation of recent work on the representation theory of general quantum groups. These form a rigid quasitensor category C to which is associated a braided group Aut(C) of braided-commutative “co-ordinate functions” analogous to the ring of functions on a group or supergroup. Every dual quasitriangular Hopf algebra A gives rise to such a braided group A. We give the example of the braided group BSL(2) in detail. We also give the rank of quantum groups in dual form and explain its connection with the partition function of simple quantum systems.
TL;DR: It was shown in this paper that the nth center of a radical ring coincides with that of its adjoint group, from which a result of Jennings is sharpened and a conjecture of his is confirmed.
Abstract: It is shown that the nth center of a radical ring coincides with that of its adjoint group, from which a result of Jennings is sharpened and a conjecture of his is confirmed.
01 May 1992
TL;DR: In this paper, Farber and Vogel proposed a localization of a ring with respect to a class of square matrices, which is called ring homomorphism and can be seen as a form of invertible matrix mapping.
Abstract: BY M. FARBERDepartment of Mathematics, Raymond and Beverly Sackler, Faculty of ExactSciences, Tel Aviv University, Ramat-Aviv, Tel Aviv 69978, IsraelAND P. VOGELDepartment of Mathematics, University of Nantes, 2 rue de Houssinie're, la F-4A072Nantes Cedex 03, France(Received 8 February 1991; revised 8 May 1991)In [1] P. Cohn suggested the construction of a localization of a ring with respectto a class of square matrices. Let us briefly recall the definitions.Let A be a ring and 2 be a set of square matrices over A. A ring homomorphismf:A->8 (S being a ring) is said to be ^.-inverting if every matrix in 2 is mapped by/ to an invertible matrix in S. A ring homomorphis a. A: -> A
TL;DR: In this paper, a ring rolling process is analyzed by the Arbitrary Lagrangian Eulerian (ALE) finite element method, which includes the overall shape of the formed ring, the time histories of roll separating force and driving torque, distribution of the normal pressure on the ring-roll interface as well as the distribution of effective stresses in a formed ring.
Abstract: In this paper, a ring rolling process is analysed by the Arbitrary Lagrangian Eulerian (ALE) finite element method. Phenomena associated with the process, such as large deformations, elastoplastic material behaviour and the friction on the interface, are included in the analysis. Special modelling on driven, idle and guide rolls is given. Results which include the overall shape of the formed ring, the time histories of roll separating force and driving torque, the distribution of the normal pressure on the ring–roll interface as well as the distribution of effective stresses in the formed ring, are also presented.
TL;DR: In this article, the authors studied functors between categories of modules graded by G-sets, and obtained generalizations of Cohen-Montgomery duality Theorems by categorical methods.
Abstract: Let G be a group, R a G-graded ring and X a right G-set. We study functors between categories of modules graded by G-sets, continuing the work of [M]. As an application we obtain generalizations of Cohen-Montgomery Duality Theorems by categorical methods. Then we study when some functors introduced in [M] (which generalize some functors ocurring in [D1], [D2] and [NRV]) are separable. Finally we obtain an application to the study of the weak dimension of a group graded ring.
01 Apr 1992
TL;DR: If G is a subgroup of finite index in the multiplicative group of an infinite field K, then G−G=K as mentioned in this paper, and the same result holds for various rings.
Abstract: If G is a subgroup of finite index in the multiplicative group of an infinite field K then G−G=K. Similar results hold for various rings
TL;DR: In this paper, the authors present a simplification of Green and Zimmermann Huisgen's theorem for left artinian rings with radical cubed zero, which they used to prove the finitistic dimension conjecture.
Abstract: The most frequently quoted finitistic dimension conjecture states that every finite dimensional algebra (or more generally, left artinian ring) has a finitely generated module of largest finite projective dimension. The historical origins of this conjecture are outlined in [7] where the conjecture was verified for monomial algebras. (Another proof was presented in [9].) So far the other most significant contribution regarding the finitistic dimension conjecture is Green and Zimmermann Huisgen's theorem establishing it for left artinian rings with radical cubed zero. In this paper we present a simplification of their proof. The methods used also yield a simple proof of the module theoretic version [3, Corollary 4.21.1] of an old theorem of Fossum, Griffith and Reiten, and verifications of the conjecture for one---sided serial rings and some rings related to, but not covered by, the results of [8]. In each case involving finitistic dimension the basic method used is a reduction of the problem to a ring whose simple modules all have infinite projective dimension. We shall use the terminology R Mod and R mod for the categories of arbitrary and, respectively, finitely generated left R-modules. Similarly, direct summands of arbitrary direct sums and, respectively, finite direct sums of copies of
TL;DR: In this article, the authors discuss two topics related with combinatorial study of canonical modules of the Stanley-Reisner ring: (i) some linear inequalities on the number of faces of a matroid complex and (ii) a formula to compute the Cohen-Macaulay type of the SRC ring of a finite distributive lattice.
Abstract: We discuss two topics related with combinatorial study of canonical modules of Stanley-Reisner rings, viz., (i) some linear inequalities on the number of faces of a matroid complex and (ii) a formula to compute the Cohen-Macaulay type of the Stanley-Reisner ring of a finite distributive lattice