Multiplication product formula for associated homogeneous distributions on R

The set of associated homogeneous distributions (AHDs) on R, ??^′(R), consists of the distributional analogues of power‐log functions with domain in R. This set contains the majority of the (one‐dimensional) distributions one typically encounters in physics applications. The recent work done by the author showed that the set ??^′(R) admits a closed convolution structure (??^′(R), *). By combining this structure with the generalized convolution theorem, a distributional multiplication product was defined, resulting in also a closed multiplication structure (??^′(R), .). In this paper, the general multiplication product formula for this structure is derived. Multiplication of AHDs on R is associative, except for critical triple products. These critical products are shown to be non‐associative in a simple and interesting way. The non‐associativity is necessary and sufficient to circumvent Schwartz's impossibility theorem on the multiplication of distributions. Copyright © 2011 John Wiley & Sons, Ltd.     

The convolution and multiplication of one‐dimensional associated homogeneous distributions

The set of associated homogeneous distributions (AHDs) with support in R is an important subset of the tempered distributions because it contains the majority of the (one‐dimensional) distributions typically encountered in physics applications (including the δ distribution). In a previous work of the author, a convolution and multiplication product for AHDs on R was defined and fully investigated. The aim of this paper is to give an easy introduction to these new distributional products. The constructed algebras are internal to Schwartz’ theory of distributions and, when one restricts to AHDs, provide a simple alternative for any of the larger generalized function algebras, currently used in non‐linear models. Our approach belongs to the same class as certain methods of renormalization, used in quantum field theory, and are known in the distributional literature as multi‐valued methods. Products of AHDs on R, based on this definition, are generally multi‐valued only at critical degrees of homogeneity. Unlike other definitions proposed in this class, the multi‐valuedness of our products is canonical in the sense that it involves at most one arbitrary constant. A selection of results of (one‐dimensional) distributional convolution and multiplication products are given, with some of them justifying certain distributional products used in quantum field theory. Copyright © 2012 John Wiley & Sons, Ltd.     

The convolution of associated homogeneous distributions on R–Part II

This is the second in a series of two papers in which we construct a convolution product for the set ?′ (R) of associated homogeneous distributions (AHDs) with support in R. In Part I we showed that if f ^a and g ^b are AHDs with degrees of homogeneity a ? 1 and b ? 1, the convolution f ^a * g ^b exists as an AHD, if the resulting degree of homogeneity a + b?1 ? N. In this article, we develop a functional extension process, based on the Hahn–Banach theorem, to give a meaning to the convolution product of two AHDs of degrees a ? 1 and b ? 1, in the critical case that a + b ? 1 ∈ N. With respect to this construction, the structure (?′(R), *) is shown to be closed.     

Structure theorems for associated homogeneous distributions based on the line

Associated homogeneous distributions (AHDs) with support in the line R are the distributional generalizations of one‐dimensional power‐log functions. In this paper, we derive a number of practical structure theorems for AHDs based on R and being complex analytic with respect to their degree of homogeneity in some region of the complex plane. Each theorem gives a representation that is designed to have a distinct advantage for calculating either convolution products, multiplication products, generalized derivatives and primitives, Fourier transforms or Hilbert transforms of AHDs. Copyright © 2008 John Wiley & Sons, Ltd.     

On self-adjoint subloops of moufang loops

In this paper we give a combinatorial rule to compute the composition of two convolution products of endomorphisms of a free associative algebra and deduce the construction of a subalgebra of QB _n (the group algebra of Hyperoctahedral group) which contains the descent algebra X#?. We also deduce a proof of the multiplication rule in the algebra ∑QB _n- Finally, we generalize this construction to other wreath products of symmetric groups by abelian groups.     

Actions of monoidal categories and generalized Hopf smash products

Let R be a k-algebra, and a monoidal category. Assume given the structure of a -category on the category of left R-modules; that is, the monoidal category is assumed to act on the category by a coherently associative bifunctor . We assume that this bifunctor is right exact in its right argument. In this setup we show that every algebra A (respectively coalgebra C) in gives rise to an R-ring AR (respectively an R-coring CR) whose modules (respectively comodules) are the A-modules (respectively C-comodules) within the category . We show that this very general scheme for constructing (co)associative (co)rings gives conceptual explanations for the double of a quasi-Hopf algebra as well as certain doubles of Hopf algebras in braided categories, each time avoiding ad hoc computations showing associativity.     

On a minimal set of generators for the invariants of 3 × 3 matrices

Let K be a field of characteristic p > 0, let L be a restricted Lie algebra and let R be an associative K-algebra. It is shown that the various constructions in the literature of crossed product of R with u(L) are equivalent. We calculate explicit formulae relating the parameters involved and obtain a formula which hints at a noncommutative version of the Bell polynomials.     

Hypergroups and binary relations

The paper deals with a binary relation R on a set H, where the Rosenberg partial hypergroupoid H _R is a hypergroup. It proves that if H _R is a hypergroup, S is an extension of R contained in the transitive closure of R and S ², then H _S is also a hypergroup. Corollaries for various extensions of R, the union, intersection and product constructions being employed, are then proved. If H _R and H _S are mutually associative hypergroups then is proven to be a hypergroup. Lastly, a tree and an iterative sequence of hyperoperations where k = 1, 2, ...) on its vertices are considered. A bound on the diameter of is given for each k such that is associative. Received December 18, 1998; accepted in final form February 8, 2000.     

On the operations of sequences in rings and binomial type sequences

Given a commutative ring with identity R, many different and interesting operations can be defined over the set \(H_R\) of sequences of elements in R. These operations can also give \(H_R\) the structure of a ring. We study some of these operations, focusing on the binomial convolution product and the operation induced by the composition of exponential generating functions. We provide new relations between these operations and their invertible elements. We also study automorphisms of the Hurwitz series ring, highlighting that some well-known transforms of sequences (such as the Stirling transform) are special cases of these automorphisms. Moreover, we introduce a novel isomorphism between \(H_R\) equipped with the componentwise sum and the set of the sequences starting with 1 equipped with the binomial convolution product. Finally, thanks to this isomorphism, we find a new method for characterizing and generating all the binomial type sequences.

     

Algebraic group actions on noncommutative spectra

Let G be an affine algebraic group and let R be an associative algebra with a rational action of G by algebra automorphisms. We study the induced G-action on the set Spec R of all prime ideals of R, viewed as a topological space with the Jacobson–Zariski topology, and on the subspace Rat R ⊆ Spec R consisting of all rational ideals of R. Here, a prime ideal P of R is said to be rational if the extended centroid is equal to the base field. Our results generalize the work of Mœglin and Rentschler and of Vonessen to arbitrary associative algebras while also simplifying some of the earlier proofs. The map P ↦ ⋂ _{g ∈ G} g.P gives a surjection from Spec R onto the set G-Spec R of all G-prime ideals of R. The fibers of this map yield the so-called G-stratification of Spec R which has played a central role in the recent investigation of algebraic quantum groups, in particular, in the work of Goodearl and Letzter. We describe the G-strata of Spec R in terms of certain commutative spectra. Furthermore, we show that if a rational ideal P is locally closed in Spec R then the orbit G.P is locally closed in Rat R. This generalizes a standard result on G-varieties. Finally, we discuss the situation where G-Spec R is a finite set. Research supported in part by NSA Grant H98230-07-1-0008.     

On endomorphisms of degree two

LetR be a commutative ring, Δ∈R and letRΔ be the set of conjugacy classes ofR-module endomorphismsf satisfyingf ∘ f = Δ·id. Using a certain subspace of the tensor product of two endomorphisms a commutative and associative product on Rx0394; can be defined. ForR = ℤ a generalization of the composition of quadratic forms arises as a special case.     

EXCHANGE MORITA RINGS

In this paper we characterize the largest exchange ideal of a ring R as the set of those elements x ∈ R such that the local ring of R at x is an exchange ring. We use this result to prove that if R and S are two rings for which there is a quasi-acceptable Morita context, then R is an exchange ring if and only if S is an exchange ring, extending an analogue result given previously by Ara and the second and third authors for idempotent rings. We introduce the notion of exchange associative pair and obtain some results connecting the exchange property and the possibility of lifting idempotents modulo left ideals. In particular we obtain that in any exchange ring, orthogonal von Neumann regular elements can be lifted modulo any one-sided ideal.     

A note on strongly π-regular rings

Let R be an associative ring with unit and let N(R) denote the set of nilpotent elements of R. R is said to be stronglyπ-regular if for each x∈R, there exist a positive integer n and an element y∈R such that x ⁿ=x ^{n +1} y and xy=yx. R is said to be periodic if for each x∈R there are integers m,n≥ 1 such that m≠n and x ^m=x ⁿ. Assume that the idempotents in R are central. It is shown in this paper that R is a strongly π-regular ring if and only if N(R) coincides with the Jacobson radical of R and R/N(R) is regular. Some similar conditions for periodic rings are also obtained. This revised version was published online in June 2006 with corrections to the Cover Date.     

On (∈, ∈ ∨ q)‐fuzzy filters of R0‐algebras

In this paper, we introduce the notions of (∈, ∈ ∨ q)‐fuzzy filters and (∈, ∈ ∨ q)‐fuzzy Boolean (implicative) filters in R₀‐algebras and investigate some of their related properties. Some characterization theorems of these generalized fuzzy filters are derived. In particular, we prove that a fuzzy set in R₀‐algebras is an (∈, ∈ ∨ q)‐fuzzy Boolean filter if and only if it is an (∈, ∈ ∨ q)‐fuzzy implicative filter. Finally, we consider the concepts of implication‐based fuzzy Boolean (implicative) filters of R₀‐algebras (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Extending Sets of Idempotents to Ring Extensions

In this paper the idea of an intrinsic extension of a ring, first proposed by Faith and Utumi, is generalized and studied in its own right. For these types of ring extensions, it is shown that, with relatively mild conditions on the base ring, R, a complete set of primitive idempotents (a complete set of left triangulating idempotents, a complete set of centrally primitive idempotents) can be constructed for an intrinsic extension, T, from a corresponding set in the base ring R. Examples and applications are given for rings that occur in functional analysis and group ring theory.     

WATTS THEOREMS FOR ASSOCIATIVE RINGS

Let R be an associative ring. In this paper we consider the category CMod-R of right R-modules M such that M ? Hom _R (R, M) and the category DMod-R of right R-modules M such that M ? _R R ? M. Given two associative rings R and R′, we study the functors F : CMod-R → CMod-R′ that can be written as Hom _R (P, ?) and the functors G : DMod-R → DMod-R′ that can be written as – ? _R Q and we give some results that extend the known Watts theorems for rings with identity to associative rings that need not be unital.     

Remarks on the r and Δ convolutions

In this paper we study the properties of the r–deformation introduced in [B1]. We observe that the associated convolution coming from the conditionally free convolution is associative only for r = 1 and r = 0. We give the realization of some r–Gaussian random variables and obtain Haagerup–Pisier–Buchholz type inequalities. We also study another convolution defined with the use of the r–deformation through a moment–cumulant formula [KY1] and show that it is associative and in general not positive. Partially sponsored with KBN grant no 2P03A00723 and RTN HPRN-CT-2002-00279.     

On derivations of finite index chain conditions,dimensions and radicals

Letd be a derivation of an associative ringR, and letM be a leftR-module withd-derivationD of finite index. It is shown thatM satisfies any of a class of conditions (including ACC, DCC, uniform, Gabriel, Krull dimension) if and only if it satisfies the same condition with respect toD-invariant submodules. If in addition 1/w! ∈R, wherew denotes the index ofd, thenD-simpleR-modules are completely reducible. Relationships between the Jacobson and theD-invariant Jacobson radicals ofM are investigated.     

Positive Derivations on Archimedean Almost f-Rings

It is shown by P. Colville, G. Davis and K. Keimel that if R is an Archimedean f-ring then a positive group endomorphism D on R is a derivation if and only if the range of D is contained in N(R) and the kernel of D contains R ², where N(R) is the set of all nilpotent elements in R and R ² is the set of all products uv in R. The main objective of this paper is to establish the result corresponding to the Colville–Davis–Keimel theorem in the almost f-ring case. The result obtained in this regard is that if D is a positive derivation in an Archimedean almost f-ring, then the range of D is contained in N(R) and the kernel of D contains R ³, where R ³ is the set of all products uvw in R. Examples are produced showing that, contrary to the f-ring case, the converse is in general false and the third power is the best possible.