Showing papers on "Ring (mathematics) published in 1979"

Algebraic number theory

Abstract: The origins of algebraic number theory.- I Algebraic Methods.- 1 Algebraic background.- 1.1 Rings and fields.- 1.2 Factorization of polynomials.- 1.3 Field extensions.- 1.4 Symmetric polynomials.- 1.5 Modules.- 1.6 Free abelian groups.- 2 Algebraic numbers.- 2.1 Algebraic numbers.- 2.2 Conjugates and discriminants.- 2.3 Algebraic integers.- 2.4 Integral bases.- 2.5 Norms and traces.- 3 Quadratic and cyclotomic fields.- 3.1 Quadratic fields.- 3.2 Cyclotomic fields.- 4 Factorization into irreducibles.- 4.1 Historical background.- 4.2 Trivial factorizations.- 4.3 Factorization into irreducibles.- 4.4 Examples of non-unique factorization into irreducibles.- 4.5 Prime factorization.- 4.6 Euclidean domains.- 4.7 Euclidean quadratic fields.- 4.8 Consequences of unique factorization.- 4.9 The Ramanujan-Nagell theorem.- 5 Ideals.- 5.1 Historical background.- 5.2 Prime factorization of ideals.- 5.3 The norm of an ideal.- II Geometric Methods.- 6 Lattices.- 6.1 Lattices.- 6.2 The quotient torus.- 7 Minkowski's theorem.- 7.1 Minkowski's theorem.- 7.2 The two-squares theorem.- 7.3 The four-squares theorem.- 8 Geometric representation of algebraic numbers.- 8.1 The space Lst.- 9 Class-group and class-number.- 9.1 The class-group.- 9.2 An existence theorem.- 9.3 Finiteness of the class-group.- 9.4 How to make an ideal principal.- 9.5 Unique factorization of elements in an extension ring.- III Number-Theoretic Applications.- 10 Computational methods.- 10.1 Factorization of a rational prime.- 10.2 Minkowski's constants.- 10.3 Some class-number calculations.- 10.4 Tables.- 11 Fermat's Last Theorem.- 11.1 Some history.- 11.2 Elementary considerations.- 11.3 Kummer's lemma.- 11.4 Kummer's Theorem.- 11.5 Regular primes.- 12 Dirichlet's Units Theorem.- 12.1 Introduction.- 12.2 Logarithmic space.- 12.3 Embedding the unit group in logarithmic space.- 12.4 Dirichlet's theorem.- Appendix 1 Quadratic Residues.- A.3 Quadratic Residues.- Appendix 2 Valuations.- References.

Tree -ring skeleton plotting by computer

Abstract: Skeleton plotting is an established manual technique for representing the relative narrowness of tree rings in a single radius. These plots can be used as a visual aid to crossdating. This paper describes a method for deriving these plots by computer. The method uses a low -pass digital filter, running means, and standard deviations of ring

On BCH codes over arbitrary integer tings (Corresp.)

Abstract: Bose-Chadhuri-Hocquenghem (BCH) codes with symbols from an arbitrary finite integer ring are derived in terms of their generator polynomials. Tile derivation is based on the factorization of x^{n}-1 over the unit ring of an appropriate extension of the Finite integer ring. The construction is thus shown to be similar to that for BCH codes over finite fields.

On the vertices of irreducible modules

Abstract: The same result holds if D is a vertex of an irreducible FG-module Me B. This second assertion generalizes some known results: For p-solvable groups G, it has been shown by Hamernik and Michler [10], that for any irreducible module M in a block B of FG, a vertex of M contains-up to conjugation-the center of a defect group of B. In [13], Michler has shown that all irreducible modules in a block with cyclic defect group A have A as a vertex. A further result in this context was given by Landrock and Michler ([11], Theorem 3.7) as a by-product of their analysis of the principal 2-blocks of the smallest Janko group and the groups of Ree-type. The proof here uses properties of the Green functor to reduce to the case D < G. Then one has to study direct summands of a module induced from a normal subgroup. This is done by inspection of the ring of endomorphisms (Sections 1 and 2). Using these results, the existence of certain B-pairs (see Definition 0.3 below) is shown in Section 3. The results mentioned at the beginning follow then from work done by Brauer and Olsson. The techniques used in the first sections yield a criterion for an ordinary irreducible character X to be of height 0: If F is algebraically closed and M an R-module affording X, then the character X belonging to the p-block B is of height 0 if and only if a vertex of M is a defect group of B and p does not divide the R-rank of a source of M. The proof is given in Section 4.

The center of the ring of 3×3 generic matrices

Abstract: Following Procesi, the center of the division ring of generic matrices over a field F is described as the fixed field of the symmetric group acting on a purely transcendental extension of F. For 3×3 matrices, the center is shown to be purely transcendental over F. In characteristic zero this is equivalent to saying that the field of simultaneous rational invariants of 3×3 matrices over F is a purely transcendental extension field of F.

Rings whose right modules are direct sums of indecomposable modules

Abstract: It is shown that, given a module M over a ring with 1, every direct product of copies of M is a direct sum of modules with local endomorphism rings if and only if every direct sum of copies of M is algebraically compact. As a consequence, the rings whose right modules are direct sums of indecomposable modules coincide with those whose right modules are direct sums of finitely generated modules.

Invariants of finite groups generated by pseudo-reflections in positive characteristic

Abstract: Let R be a commutative ring, and let V be a finitelygenerated freei?-module. Let R[V] be a polynomial ring over R associated with V. Then a finitesubgroup G of GL(V) acts naturally on R[V]. We denote by R[V]G the ring of invariants of R[V] under the action of G. Let R=k be a fieldand suppose that \G\is a unit of k. It is known ([4],[9], [3],[8]) that k[V]G is a polynomial ring if and only if G is generated by pseudoreflectionsin GL{V). But, in the case where \G\=0 mod char{k), there are only the following results: (1) L. E. Dickson [5]; FqlTu ・・・,rB]O£cn.9) an(jFq[Tu ■-,TnfLin^ are polynomial rings, where Fq is the finitefieldof q elements. (2) M.-J. Bertin [1]; Fq[Tu ■・-, Tnfnipin-^ is a polynomial ring, where

The duality operation in the character ring of a finite Chevalley group

Abstract: It is possible (as in [4] ) to define a duality operation f —• f * in the ring of virtual characters of an arbitrary finite group with a split (B, 7V)-pair of characteristic p. Such a group arises as the fixed points under a Frobenius map of a connected reductive algebraic group, defined over a finite field [1]. This paper contains statements of several general properties of the duality map f —• f * and two related operations (see §§2 and 4). The duality map f —• f * generalizes the construction in [2] of the Steinberg character, and interacts well with the organization of the characters from the point of view of cuspidal characters (§6). It is hoped that there is also a useful interaction with the Deligne-Lusztig virtual characters R^O. Partial results have been obtained in this direction (§5). Detailed proofs will appear elsewhere.

Method of manufacturing a blow-molded container with an integral handle

Abstract: Method for forming blow molded containers with integral handles for carrying and/or pouring. A preform with an integral handle having a ring and projection is formed, supported by the ring, and expanded below the ring to form a container. The projection can be maintained spaced from the expanded container portion or can be joined thereto during or after expansion.

A right PCI ring is right Noetherian

Abstract: C. Faith and J. Cozzens have shown that a ring, whose right proper cyclic modules are injective, is either semisimple or a simple, right semihereditary, right Ore V-domain. They have posed a question as to whether such a ring is right noetherian. In this paper, an affirmative answer is given to that question. Moreover, necessary and sufficient conditions are given as to when a right PCI ring is left PCI. In [1], Faith and Cozzens proved that a ring R whose proper right cyclic modules are injective must be either semisimple or a simple, right semihereditary, right Ore V-domain. They noted that all the known examples of such rings are right noetherian and posed the question whether every ring with this property is noetherian. We will answer this question in the affirmative. More clearly, we will show: THEOREM 1. Let R be a right PCI ring, then either (a) R is semisimple, or (b) R is a simple right noetherian, right hereditary, right Ore V-domain. By using Boyle's theorem [1, Theorem 6.26] and Theorem 1, we get the following immediately: COROLLARY. R is a PCI ring, that is a right and left PCI ring, if and only if R is either semisimple or a simple noetherian hereditary Ore V-domain. C1 PROOF OF THEOREM. Faith and Cozzens have shown that a PCI ring R must be either semisimple or a simple, right semihereditary, right Ore V-domain [1, Theorems 6.13, 6.17]. Thus, we may assume R is a simple, right semihereditary, right Ore V-domain that is not semisimple. Furthermore it is known that such a ring has the property that every proper finitely presented cyclic module has a von Neumann regular endomorphism ring [1, Proposition 6.20]. Suppose that the endomorphism ring of every proper finitely presented cyclic module is semisimple. Then, every proper finitely presented cyclic module has an indecomposable decomposition. It is clear from the properties of R that the only indecomposable injective modules are the simple modules and E(RR). Thus, it follows that every proper finitely presented cyclic module is a direct sum of a semisimple module and a finite number of isomorphic copies of E(RR). Received by the editors October 5, 1978 and, in revised form, January 22, 1979. AMS (MOS) subject classifications (1970). Primary 16A48. ? 1979 American Mathematical Society 0002-9939f79f0000-0452/$02.00 This content downloaded from 207.46.13.184 on Fri, 29 Jul 2016 05:57:32 UTC All use subject to http://about.jstor.org/terms

Systems aspects of The Cambridge Ring

Abstract: The Cambridge Ring is a local communication system developed in the Computer Laboratory of the University of Cambridge. It differs in various respects from some other local communication mechanisms such as Ethernet systems (Metcalfe & Boggs, 1976), and the purpose of the present paper is to describe the way in which the properties of the ring affect the general systems aspects of its exploitation.

A canonical operad pair

Abstract: The purpose of this paper is to construct an operad ℋ∞ with the good properties of both the little convex bodies partial operad and the little cubes operad used in May's theory of E∞ ring spaces or multiplicative infinite loop spaces ((6), chapter VII). In (6) ℋ can then be used instead of and , and the theory becomes much simpler; in particular all partial operads can be replaced by genuine ones. The method used here is a modification of that which May suggests on (6), page 170, but cannot carry out.

Direct product decomposition of theories of modules

Abstract: This paper is mainly concerned with describing complete theories of modules by decomposing them (up to elementary equivalence) into direct products of simpler modules. In §1, I give a decomposition theorem which works for arbitrary direct product theories T. Given such a T, I define T-indecomposable structures and show that every model of T is elementarily equivalent to a direct product of T-indecomposable models of T. In §2, I show that if R is a commutative ring then every R-module is elementarily equivalent to ΠMM where M ranges over the maximal ideals of R and M is the localization of at M. This is applied to prove that if R is a commutative von Neumann regular ring and TR is the theory of R-modules then the TR-indecomposables are precisely the cyclic modules of the form R/M where M is a maximal ideal. In §3, I use the decomposition established in §2 to characterize the ω1-categorical and ω-stable modules over a countable commutative von Neumann regular ring and the superstable modules over a commutative von Neumann regular ring of arbitrary cardinality. In the process, I also prove several general characterizations of ω-stable and superstable modules; e.g., if R is any countable ring, then an R-moduIe is ω-stable if and only if every R-module elementarily equivalent to it is equationally compact.

On the Embedding into a Ring of an Archimedean ι-Group

Abstract: We shall prove the following about the “ringification” ρA of [2] and [5] of an archimedean l-group A: (a) Any “minimal ring” containing A is ρA; (b) A ↦ ρA is a reflector; (c) ρA need not be laterally complete when A is. These constitute the solutions to the problems posed in [2] by Paul Conrad. 1. The embedding into a ring. Let be the category which has objects archimedean l-groups A with distinguished positive weak unit eA , and morphisms l-group homomorphisms h: A → B with h(eA) = eB . Let be the category with objects archimedean f-rings R with identity 1 R which is a weak unit, and morphisms l-ring homomorphisms h: R → S with h(l R ) = 1 S .

New results in realization theory for linear time-varying analytic systems

Abstract: The realization of linear time-varying systems specified by an analytic weighting pattern is approached in a novel manner using an algebraic framework defined over the ring of analytic functions. Realizations are given by a state representation consisting of a first-order vector differential equation and an output equation, both with analytic coefficients. Various new criteria for realizability are derived, including conditions given in terms of the finiteness of modules over the ring of analytic functions generated by the elementary rows or columns of a (generalized) Hankel matrix. These results are related to local criteria for realizability specified in terms of the rank of matrix functions, as developed in the work of Silverman and Meadows [5], [8], [9] and Kalman [7]. It is shown that the construction of minimal realizations reduces to the problem of computing a basis for a finite free module defined over the ring of analytic functions. A minimal realization algorithm is then derived using a constructive procedure for computing bases for finite free modules over a Bezout domain. The Silverman-Meadows realization algorithm [5] is a special case of the procedure given here. In the last part of the paper, the realization algorithm is applied to the problem of system reduction.

τ-rings and wreath product representations

Abstract: Wreath products and representations.- Tensor power constructions.- ?-Rings.- Operations and the free ?-ring S.- Computations related to R(?k).- ?-Rings, adams operations and plethysms.- Computations related to R(?k ).- Filtered ?-rings.- Problems and analogues.

Kinetic theory of ion ring compression

Abstract: A general kinetic theory is developed for the adiabatic compression of ion rings, and associated plasma, in an external magnetic mirror field Be(t). The single particle distribution function is shown to be given by f (H,Pϑ,t) =F (I,Pϑ) during axisymmetric compression in the absence of instabilities and under conditions where the particle motion is ergodic in the poloidal (r,z) plane. Here, H and Pϑ are the energy and canonical momentum of a ring ion; I (H,Pϑ) is an adiabatic invariant involving an average over the region of the poloidal plane accessible to a particle with constants H and Pϑ; and F is an invariant function. A complete description of ring compression is obtained from F, which gives the ring current density Jbϑ, from an appropriate constitutive relation for the plasma response, which gives the plasma current density Jpϑ, and from Ampere’s law. The theory is valid for rings of arbitrary aspect ratio, the particle orbits are essentially unrestricted, and account is taken of ring particle evapo...

Definable principal congruences in varieties of groups and rings

Abstract: In [lJ Baldwin and Berman showed that for varieties ~ with DPC (definable principal congruences) certain results of Taylor concerning residually small varieties could be sharpened. Their question as to whether every variety generated by a finite algebra has DPC was answered in the negative in [2]; however the question remained open for varieties with permutable congruences. The study of DPC became even more interesting when McKenzie [4] proved that this property could be used, in certain cases (such as a variety generated by a para-primal algebra), to give an easy proof of the finite axiomatizablity of the variety. McKenzie then showed that among lattices only the distributive varieties have DPC, and states that the question of whether varieties generated by a finite group or ring have DPC is open. In the first section we point out that a variety ~ has DPC iff the free algebra on countably many generators in ~ has SDPC (strongly definable principal congruences), hence a variety generated by a class Y{ of algebras has DPC iff the quasi-variety generated by 5g has DPC. In the second section a finite ring R is constructed such that the variety generated by R does not have DPC. In the third section we prove that if the variety generated by a finite group G has DPC then G must be nilpotent; on the other hand if G is nilpotent class 2 and finite then indeed it generates a variety with DPC. It follows that the properties of having DPC and being finitely axiomatizable are independent for quasi-varieties generated by a finite group. Finally Baldwin's theorem (3) that the variety of all groups of exponent 3 has DPC is shown to be best possible for Burnside varieties.

Maintenance of Ring Communication Systems

Abstract: This note presents a simple way of localizing errors and failures in ring communication systems. The method is also applicable to some other types of networks.