Finite Generation of Units in Alternative Loop Rings

Let L be an RA loop, that is, a loop whose loop ring in any characteristic is an alternative, but not associative, ring. Suppose that L is finite and that any noncommutative division algebra appearing as a simple component in the Wedderburn decomposition of Q L is the classical Cayley–Dickson algebra over Q. Then the unit loop of the alternative loop ring Z L of L over the ring of rational integers is finitely generated.     

Central units in salternative loop rings

Let L be an RA loop, that is, a loop whose loop ring in any characteristic is an alternative, but not associative, ring. We show that every central unit in the integral loop ring ZL is the product ℓμ₀ of an element ℓ ∈ L and a loop ring element μ₀ whose support is in the torsion subloop of L and use this result to determine when all central units of ZL are trivial. Received: 8 October 2004     

Faulkner geometry

Axioms are presented for a Barbilian geometry of dimension n2 over a ring for which ab=1 implies ba=1. It is shown that any Faulkner geometry of dimension n3 is coordinatized by a unique associative two-sided units ring R and that the group generated by all transvections is a group of Steinberg type over R.     

The formal theory of Hopf algebras Part II: The case of Hopf algebras

Abstract

The category Hopf_R of Hopf algebras over a commutative unital ring R is analyzed with respect to its categorical properties. The main results are: (1) For every ring R the category Hopf_R is locally presentable, it is coreflective in the category of bialgebras over R, over every R-algebra there exists a cofree Hopf algebra. (2) If, in addition, R is absoluty flat, then Hopf_R is reflective in the category of bialgebras as well, and there exists a free Hopf algebra over every R-coalgebra. Similar results are obtained for relevant subcategories of Hopf_R. Moreover it is shown that, for every commutative unital ring R, the so-called “dual algebra functor” has a left adjoint and that, more generally, universal measuring coalgebras exist.     

Differential rings and ore extensions: Brown-McCoy rings

We consider here a ringK, a derivationD ofK and the differential polynomial ringR=K[X;D]. The ringK is said to be a Brown-McCoy ring if the prime radical coincides with the Brown-McCoy radical in every homomorphic image ofK. AD-Brown-McCoy ring is defined in a similar way. We prove the following conditions are equivalent: (i)K is aD-Brown-McCoy ring; (ii)R is a Brown-McCoy ring and for every maximal idealM ofR,K/(MνK) is aD-simple ring with 1. In addition, we give some applications and examples on the study of the transfer of the property of being a Brown-McCoy ring betweenK andR. Further, we study the relation between the prime and theD-prime ideals of a differential intermediate extension of a liberal extension. This paper was supported by a fellowship awarded by Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq), Brazil.     

The module of derivations of regular local rings

It is shown that if A is a ring and A is a regular local k-algebra such that Derk(A) is finitely generated, then it is free under some conditions.     

Symmetric functions and the centre of the Ringel-Hall algebra of a cyclic quiver

We obtain explicit generators for the centre of the Ringel-Hall algebra of a cyclic quiver and define a canonical algebra monomorphism from Macdonald's ring of symmetric functions to the centre, which furthermore respects the comultiplication and the symmetric bilinear form. Dedicated to Claus Michael Ringel on the occasion of his 60th birthday     

Positivity and Convexity in Rings of Fractions

Given a commutative ring A equipped with a preordering A⁺ (in the most general sense, see below), we look for a fractional ring extension (= “ring of quotients” in the sense of Lambek et al. [L]) as big as possible such that A⁺ extends to a preordering R⁺ of R (i.e. with A ∩ R⁺ = A⁺) in a natural way. We then ask for subextensions A ⊂ B of A ⊂ R such that A is convex in B with respect to B⁺ : = B ∩ R⁺. Supported by DFG. A short form of this article has been delivered at the conference Carthapos 2006 at Carthago (Tunisia).     

Decomposing thick subcategories of the stable module category

Let be the stable category of finitely generated modular representations of a finite group G over a field k. We prove a Krull-Remak-Schmidt theorem for thick subcategories of . It is shown that every thick tensor-ideal of (i.e. a thick subcategory which is a tensor ideal) has a (usually infinite) unique decomposition into indecomposable thick tensor-ideals. This decomposition follows from a decomposition of the corresponding idempotent kG-module into indecomposable modules. If is the thick tensor-ideal corresponding to a closed homogeneous subvariety W of the maximal ideal spectrum of the cohomology ring , then the decomposition of reflects the decomposition of W into connected components. Received: 27 April 1998 / In revised form: 16 July 1998     

Der Satz von Tits für PGL2(R), R ein kommutativer Ring vom stabilen Rang 2

Certain permutation groups on sets with distance relation are characterized as groups of projectivities PGL₂(R) on the projective line over a commutative ring R of stable rank 2, thus generalizing a classical result of Tits where R is a field.

     

On zero-divisors of near-rings of polynomials

In this paper, we are interested to study zero-divisor properties of a 0-symmetric nearring of polynomials R₀[x], when R is a commutative ring. We show that for a reduced ring R, the set of all zero-divisors of R₀[x], namely Z(R₀[x]), is an ideal of R₀[x] if and only if Z(R) is an ideal of R and R has Property (A). For a non-reduced ring R, it is shown that Z(R₀[x]) is an ideal of Z(R₀[x]) if and only if ann_R({a, b}) ∩ N i?(R) ≠ 0, for each a, b ∈ Z(R). We also investigate the interplay between the algebraic properties of a 0-symmetric nearring of polynomials R₀[x] and the graph-theoretic properties of its zero-divisor graph. The undirected zero-divisor graph of R₀[x] is the graph Γ(R₀[x]) such that the vertices of Γ(R₀[x]) are all the non-zero zero-divisors of R₀[x] and two distinct vertices f and g are connected by an edge if and only if f ? g = 0 or g ? f = 0. Among other results, we give a complete characterization of the possible diameters of Γ(R₀[x]) in terms of the ideals of R. These results are somewhat surprising since, in contrast to the polynomial ring case, the near-ring of polynomials has substitution for its “multiplication” operation.     

Genus polynomials and crosscap‐number polynomials for ring‐like graphs

An H‐linear graph is obtained by transforming a collection of copies of a fixed graph H into a chain. An H‐ring‐like graph is formed by binding the two end‐copies of H in such a chain to each other. Genus polynomials have been calculated for bindings of several kinds. In this paper, we substantially generalize the rules for constructing sequences of H‐ring‐like graphs from sequences of H‐linear graphs, and we give a general method for obtaining a recursion for the genus polynomials of the graphs in a sequence of ring‐like graphs. We use Chebyshev polynomials to obtain explicit formulas for the genus polynomials of several such sequences. We also give methods for obtaining recursions for partial genus polynomials and for crosscap‐number polynomials of a bar‐ring of a sequence of disjoint graphs.     

A note on some chain conditions

We answer a question raised by Othman Echi: Is an E ₁ (resp., a C ₁) ring an E (resp., a C) ring? We construct a C ₁ (thus E ₁) ring which is not an E ₂ (thus not a C ₂) ring. Received: 11 June 2007     

Motivic Serre invariants of curves

In this article, we generalize the theory of motivic integration on formal schemes topologically of finite type and the notion of motivic Serre invariant, to a relative point of view. We compute the relative motivic Serre invariant for curves defined over the field of fractions of a complete discrete valuation ring R of equicharacteristic zero. One aim of this study is to understand the behavior of motivic Serre invariants under ramified extension of the ring R. Thanks to our constructions, we obtain, in particular, an expression for the generating power series, whose coefficients are the motivic Serre invariant associated to a curve, computed on a tower of ramified extensions of R. We give an interpretation of this series in terms of the motivic zeta function of Denef and Loeser.     

A Complete Solution to Spectrum Problem for Five‐Vertex Graphs with Application to Traffic Grooming in Optical Networks

A G‐design of order n is a decomposition of the complete graph on n vertices into edge‐disjoint subgraphs isomorphic to G. Grooming uniform all‐to‐all traffic in optical ring networks with grooming ratio C requires the determination of graph decompositions of the complete graph on n vertices into subgraphs each having at most C edges. The drop cost of such a grooming is the total number of vertices of nonzero degree in these subgraphs, and the grooming is optimal when the drop cost is minimum. The existence spectrum problem of G‐designs for five‐vertex graphs is a long standing problem posed by Bermond, Huang, Rosa and Sotteau in 1980, which is closely related to traffic groomings in optical networks. Although considerable progress has been made over the past 30 years, the existence problems for such G‐designs and their related traffic groomings in optical networks are far from complete. In this paper, we first give a complete solution to this spectrum problem for five‐vertex graphs by eliminating all the undetermined possible exceptions. Then, we determine almost completely the minimum drop cost of 8‐groomings for all orders n by reducing the 37 possible exceptions to 8. Finally, we show the minimum possible drop cost of 9‐groomings for all orders n is realizable with 14 exceptions and 12 possible exceptions.     

The Zero Divisor Graph of 2 × 2 Matrices Over a Field

A zero divisor graph, Γ(R), is formed from a ring R by having each element of Z(R) \ {0} to be a vertex in the graph and having two vertices u and v adjacent if the corresponding elements from the ring are nonequal and have product equal to zero. In this paper, the structure of the zero-divisor graph of 2 × 2 matrices over a field, Γ(M2(F)), are completely determined.     

Families of cuts with the MFMC-property

A family ℱ of cuts of an undirected graphG=(V, E) is known to have the weak MFMC-property if (i) ℱ is the set ofT-cuts for someT⊆V with |T| even, or (ii) ℱ is the set of two-commodity cuts ofG, i.e. cuts separating any two distinguished pairs of vertices ofG, or (iii) ℱ is the set of cuts induced (in a sense) by a ring of subsets of a setT⊆V. In the present work we consider a large class of families of cuts of complete graphs and prove that a family from this class has the MFMC-property if and only if it is one of (i), (ii), (iii).     

Restricted Elasticity and Rings of Integer-Valued Polynomials Determined by Finite Subsets

Let D be an integral domain such that Int(D) ≠ K[X] where K is the quotient field of D. There is no known example of such a D so that Int(D) has finite elasticity. If E is a finite nonempty subset of D, then it is known that Int(E, D) = {f(X) ∈ K[X] | f(e) ∈ D for all e ∈ E} is not atomic. In this note, we restrict the notion of elasticity so that it is applicable to nonatomic domains. For each real number r ≥ 1, we produce a ring of integer-valued polynomials with restricted elasticity r. We further show that if D is a unique factorization domain and E is finite with |E| > 1, then the restricted elasticity of Int(E, D) is infinite. Part of this work was completed while the first author was on an Academic Leave granted by the Trinity University Faculty Development Committee.     

Congruence subgroups in group rings *

For a large class of groups G a precise congruence subgroup of the group generated by the bicyclic units of the integral group ring ZG is determined. As an application an upper bound is calculated for the index in the unit group of ZG for the group generated by the Bass cyclic units and the bicyclic units.     

On the category of modules of Gorenstein dimension zero

Let R be a commutative noetherian henselian non-Gorenstein local ring of depth zero. Denote by modR the category of all finitely generated R-modules, and by the full subcategory of modR consisting of all R-modules of Gorenstein dimension zero. We prove in this paper that if contains a non-free module, then it is not precovering in modR, in particular, there exist infinitely many isomorphism classes of indecomposable R-modules of Gorenstein dimension zero.