ISRN Algebra

A New Criterion for Affineness

Jing Zhang — Wed, 20 Mar 2013 09:02:31 +0000

We show that an irreducible quasiprojective variety of dimension defined over an algebraically closed field with characteristic zero is an affine variety if and only if () = 0 and () = 0 for all , , where is any hypersurface with sufficiently large degree. A direct application is that an irreducible quasiprojective variety over is a Stein variety if it satisfies the two vanishing conditions. Here, all sheaves are algebraic.

On Cubic KU-Ideals of KU-Algebras

Naveed Yaqoob, Samy M. Mostafa, and Moin A. Ansari — Tue, 26 Feb 2013 16:36:32 +0000

We introduce the notion of cubic KU-ideals of KU-algebras and several results are presented in this regard. The image, preimage, and cartesian product of cubic KU-ideals of KU-algebras are defined.

Idempotent Elements of the Endomorphism Semiring of a Finite Chain

Ivan Trendafilov and Dimitrinka Vladeva — Tue, 26 Feb 2013 08:05:00 +0000

Idempotents yield much insight in the structure of finite semigroups and semirings. In this article, we obtain some results on (multiplicatively) idempotents of the endomorphism semiring of a finite chain. We prove that the set of all idempotents with certain fixed points is a semiring and find its order. We further show that this semiring is an ideal in a well-known semiring. The construction of an equivalence relation such that any equivalence class contains just one idempotent is proposed. In our main result we prove that such an equivalence class is a semiring and find its order. We prove that the set of all idempotents with certain jump points is a semiring.

Semientwining Structures and Their Applications

Florin F. Nichita, Deepak Parashar, and Bartosz Zieliński — Tue, 19 Feb 2013 08:58:38 +0000

Semientwining structures are proposed as concepts simpler than entwining structures, yet they are shown to have interesting applications in constructing intertwining operators and braided algebras, lifting functors, finding solutions for Yang-Baxter systems, and so forth. While for entwining structures one can associate corings, for semientwining structures one can associate comodule algebra structures where the algebra involved is a bialgebra satisfying certain properties.

A Generalization for -Cocycles

Beishang Ren and Shixun Lin — Sun, 30 Dec 2012 08:57:07 +0000

We will give generalized definitions called type II -cocycles and weak quasi-bialgebra and also show properties of type II -cocycles and some results about weak quasi-bialgebras, for instance, construct a new structure of tensor product algebra over a module algebra on weak quasi-bialgebras.

Commutativity Theorems for *-Prime Rings with Differential Identities on Jordan Ideals

A. Mamouni, L. Oukhtite, and M. Samman — Sat, 29 Dec 2012 11:29:31 +0000

In this paper we explore commutativity of -prime rings in which derivations satisfy certain differential identities on Jordan ideals. Furthermore, examples are given to demonstrate that our results cannot be extended to semiprime rings.

Powers of Commutators and Anticommutators

Stephen M. Buckley and Desmond MacHale — Mon, 24 Dec 2012 19:06:32 +0000

For , we give elementary proofs of commutativity of rings in which the identity holds for all commutators . For even , we show that the commutativity of rings satisfying such an identity is equivalent to the anticommutativity of rings satisfying the corresponding anticommutator equation.

On Generalized ()-Derivations in Semiprime Rings

Basudeb Dhara and Atanu Pattanayak — Mon, 03 Dec 2012 16:18:05 +0000

Let be a semiprime ring, a nonzero ideal of , and , two epimorphisms of . An additive mapping is generalized -derivation on if there exists a -derivation such that holds for all . In this paper, it is shown that if , then contains a nonzero central ideal of , if one of the following holds: (i) ; (ii) ; (iii) ; (iv) ; (v) for all .

Growth for Algebras Satisfying Polynomial Identities

Amitai Regev — Wed, 21 Nov 2012 14:05:59 +0000

The th codimension of a PI algebra measures how many identities of degree the algebra satisfies. Growth for PI algebras is the rate of growth of as goes to infinity. Since in most cases there is no hope in finding nice closed formula for , we study its asymptotics. We review here such results about , when is an associative PI algebra. We start with the exponential bound on then give few applications. We review some remarkable properties (integer and half integer) of the asymptotics of . The representation theory of the symmetric group is an important tool in this theory.

On Pre-Hilbert Noncommutative Jordan Algebras Satisfying ‖𝑥2‖=‖𝑥‖2

Mohamed Benslimane and Abdelhadi Moutassim — Thu, 23 Aug 2012 10:02:53 +0000

Let 𝐴 be a real or complex algebra. Assuming that a vector space 𝐴 is endowed with a pre-Hilbert norm ‖⋅‖ satisfying ‖𝑥2‖=‖𝑥‖2 for all 𝑥∈𝐴. We prove that 𝐴 is finite dimensional in the following cases. (1) 𝐴 is a real weakly alternative algebra without divisors of zero. (2) 𝐴 is a complex powers associative algebra. (3) 𝐴 is a complex flexible algebraic algebra. (4) 𝐴 is a complex Jordan algebra. In the first case 𝐴 is isomorphic to ℝ,ℂ,ℍ, or 𝕆, and 𝐴 is isomorphic to ℂ in the last three cases. These last cases permit us to show that if 𝐴 is a complex pre-Hilbert noncommutative Jordan algebra satisfying ‖𝑥2‖=‖𝑥‖2 for all 𝑥∈𝐴, then 𝐴 is finite dimensional and is isomorphic to ℂ. Moreover, we give an example of an infinite-dimensional real pre-Hilbert Jordan algebra with divisors of zero and satisfying ‖𝑥2‖=‖𝑥‖2 for all 𝑥∈𝐴.

A Gelfand Model for Weyl Groups of Type 𝐷2𝑛

José O. Araujo, Luis C. Maiarú, and Mauro Natale — Tue, 07 Aug 2012 13:35:09 +0000

A Gelfand model for a finite group 𝐺 is a complex representation of 𝐺, which is isomorphic to the direct sum of all irreducible representations of 𝐺. When 𝐺 is isomorphic to a subgroup of 𝐺𝐿𝑛(ℂ), where ℂ is the field of complex numbers, it has been proved that each 𝐺-module over ℂ is isomorphic to a 𝐺-submodule in the polynomial ring ℂ[𝑥1,…,𝑥𝑛], and taking the space of zeros of certain 𝐺-invariant operators in the Weyl algebra, a finite-dimensional 𝐺-space 𝒩𝐺 in ℂ[𝑥1,…,𝑥𝑛] can be obtained, which contains all the simple 𝐺-modules over ℂ. This type of representation has been named polynomial model. It has been proved that when 𝐺 is a Coxeter group, the polynomial model is a Gelfand model for 𝐺 if, and only if, 𝐺 has not an irreducible factor of type 𝐷2𝑛, 𝐸7, or 𝐸8. This paper presents a model of Gelfand for a Weyl group of type 𝐷2𝑛 whose construction is based on the same principles as the polynomial model.

Cogredient Standard Forms of Symmetric Matrices over Galois Rings of Odd Characteristic

Yonglin Cao — Thu, 19 Jul 2012 17:52:41 +0000

Let 𝑅=GR(𝑝𝑠,𝑝𝑠𝑚) be a Galois ring of characteristic 𝑝𝑠 and cardinality 𝑝𝑠𝑚, where 𝑠 and 𝑚 are positive integers and 𝑝 is an odd prime number. Two kinds of cogredient standard forms of symmetric matrices over 𝑅 are given, and an explicit formula to count the number of all distinct cogredient classes of symmetric matrices over 𝑅 is obtained.

On the Completely Positive and Positive Semidefinite-Preserving Cones—Part III

Andrew J. Klimas and Richard D. Hill — Mon, 02 Jul 2012 14:20:00 +0000

This paper studies the extremals and other faces of the completely positive and positive semidefinite-preserving linear transformations.

Acceptable Morita Contexts for Semigroups

Valdis Laan — Sun, 01 Jul 2012 11:17:45 +0000

This short note deals with Morita equivalence of (arbitrary) semigroups. We give a necessary and sufficient condition for a Morita context containing two semigroups S and T to induce an equivalence between the category of closed right S-acts and the category of closed right T-acts.

Codes over Graphs Derived from Quotient Rings of the Quaternion Orders

Cátia R. de O. Quilles Queiroz and Reginaldo Palazzo Júnior — Wed, 27 Jun 2012 10:14:54 +0000

We propose the construction of signal space codes over the quaternion orders from a graph associated with the arithmetic Fuchsian group Γ8. This Fuchsian group consists of the edge-pairing isometries of the regular hyperbolic polygon (fundamental region) P8, which tessellates the hyperbolic plane 𝔻2. Knowing the generators of the quaternion orders which realize the edge pairings of the polygon, the signal points of the signal constellation (geometrically uniform code) derived from the graph associated with the quotient ring of the quaternion order are determined.

Amenability of the Restricted Fourier Algebras

Massoud Amini and Marzieh Shams Yousefi — Wed, 27 Jun 2012 10:12:59 +0000

We discuss amenability of the restricted Fourier-Stieltjes algebras on inverse semigroups. We show that, for an E-unitary inverse semigroup, amenability of the restricted Fourier-Stieltjes algebra is related to the amenability of an associated Banach algebra on a Fell bundle.

Another Proof of the Faithfulness of the Lawrence-Krammer Representation of the Braid Group 𝐵𝟑

Mohammad N. Abdulrahim and Mariam Hariri — Mon, 25 Jun 2012 08:29:07 +0000

The Lawrence-Krammer representation of the braid group 𝐵𝑛 was proved to be faithful for 𝑛≥3 by Bigelow and Krammer. In our paper, we give a new proof in the case 𝑛=3 by using matrix computations. First, we prove that the representation of the braid group 𝐵3 is unitary relative to a positive definite Hermitian form. Then we show the faithfulness of the representation by specializing the indeterminates q and t to complex numbers on the unit circle rather than specializing them to real numbers as what was done by Krammer.

When Is the Complement of the Zero-Divisor Graph of a Commutative Ring Complemented?

S. Visweswaran — Sat, 16 Jun 2012 15:10:53 +0000

Let 𝑅 be a commutative ring with identity which has at least two nonzero zero-divisors. Suppose that the complement of the zero-divisor graph of 𝑅 has at least one edge. Under the above assumptions on 𝑅, it is shown in this paper that the complement of the zero-divisor graph of 𝑅 is complemented if and only if 𝑅 is isomorphic to 𝐙/3𝐙×𝐙/3𝐙 as rings. Moreover, if 𝑅 is not isomorphic to 𝐙/3𝐙×𝐙/3𝐙 as rings, then, it is shown that in the complement of the zero-divisor graph of 𝑅, either no vertex admits a complement or there are exactly two vertices which admit a complement.

On the Constructibility of Real 5th Roots of Rational Numbers with Marked Ruler and Compass

Elliot Benjamin — Tue, 05 Jun 2012 09:20:52 +0000

We demonstrate that there are infinitely many real numbers constructible by marked ruler and compass which are unique real roots of irreducible quintic polynomials over the field of rational numbers. This result can be viewed as a generalization of the historical open question of the constructibility by marked ruler and compass of real 5th roots of rational numbers. We obtain our results through marked ruler and compass constructions involving the intersection of conchoids and circles, and the application of number theoretic divisibility criteria.

On g-Semisymmetric Rings

Farahat S. Aly and Mohammed O. Al Mestady — Thu, 17 May 2012 10:19:23 +0000

We introduce right (left) g-semisymmetric ring as a new concept to generalize the well-known concept: symmetric ring. Examples are given to show that these classes of rings are distinct. They coincide under some conditions. It is shown that 𝑅 is bounded right g-semisymmetric with boundary 1 from right if and only if 𝑅 is symmetric, whenever 𝑅 is regular. It is shown that a ring 𝑅 is strongly regular if and only if 𝑅 is regular and bounded right g-semisymmetric with boundary 1 from right. For a right 𝑝.𝑝.-ring 𝑅 it is shown that 𝑅 is reduced if and only if 𝑅 is symmetric, if and only if 𝑅 is bounded right g-semisymmetric ring with boundary 1 from left, if and only if 𝑅 is IFP, if and only if 𝑅 is abelian. We prove that there is a special subring of the ring of 3×3 matrices over a ring without zero divisors which is bounded right g-semisymmetric with boundary 2 from left and boundary 2 from right. Also we show that flat left modules over bounded left g-semisymmetric ring with boundaries 1 from left and 1 from right are bounded left g-semisymmetric with boundaries 1 from left and 1 from right.

On the Existence of (𝑣,𝑘,𝜆) Difference Sets with 𝑘<1250 and 𝑘−𝜆 Is a Square

Adegoke Solomon Osifodunrin — Tue, 15 May 2012 07:52:17 +0000

We combine Turyn's self-conjugacy result, variance technique, Dillon dihedral trick, and Sylow theorem to investigate the existence of (𝑣,𝑘,𝜆) difference sets in which 𝑘−𝜆 is a square and 𝑘<1250.

Tensor Products of Noncommutative 𝐿𝑝-Spaces

Somlak Utudee — Mon, 14 May 2012 10:24:58 +0000

We consider the notion of tensor product of noncommutative 𝐿𝑝 spaces associated with finite von Neumann algebras and define the notion of tensor product of Haagerup noncommutative 𝐿𝑝 spaces associated with 𝜎-finite von Neumann algebras.

The Matrix Linear Unilateral and Bilateral Equations with Two Variables over Commutative Rings

N. S. Dzhaliuk and V. M. Petrychkovych — Tue, 08 May 2012 11:56:06 +0000

The method of solving matrix linear equations 𝐴𝑋+𝐵𝑌=𝐶 and 𝐴𝑋+𝑌𝐵=𝐶 over commutative Bezout domains by means of standard form of a pair of matrices with respect to generalized equivalence is proposed. The formulas of general solutions of such equations are deduced. The criterions of uniqueness of particular solutions of such matrix equations are established.

An Application of Iterative Pushdown Automata to Contour Words of Balls and Truncated Balls in Hyperbolic Tessellations

Maurice Margenstern — Thu, 29 Mar 2012 10:01:14 +0000

We give an application of iterated pushdown automata to contour words of balls and two other domains in infinitely many tilings of the hyperbolic plane. We also give a similar application for the tiling {5,3,4} of the hyperbolic 3D space and for the tiling {5,3,3,4} of the hyperbolic 4D space as well.

A New Proof of the Existence of Free Lie Algebras and an Application

Andrea Bonfiglioli and Roberta Fulci — Wed, 07 Mar 2012 11:26:35 +0000

The existence of free Lie algebras is usually derived as a consequence of the Poincaré-Birkhoff-Witt theorem. Moreover, in order to prove that (given a set 𝑋 and a field 𝕂 of characteristic zero) the Lie algebra ℒ(𝕂⟨𝑋⟩) of the Lie polynomials in the letters of 𝑋 (over the field 𝕂) is a free Lie algebra generated by 𝑋, all available proofs use the embedding of a Lie algebra 𝔤 into its enveloping algebra 𝒰(𝔤). The aim of this paper is to give a much simpler proof of the latter fact without the aid of the cited embedding nor of the Poincaré-Birkhoff-Witt theorem. As an application of our result and of a theorem due to Cartier (1956), we show the relationships existing between the theorem of Poincaré-Birkhoff-Witt, the theorem of Campbell-Baker-Hausdorff, and the existence of free Lie algebras.

An Algorithm for Generating a Family of Alternating Knots

Carlos Velarde, Ernesto Bribiesca, and Wendy Aguilar — Thu, 26 Jan 2012 15:08:58 +0000

An algorithm for generating a family of alternating knots (which are described by means of a chain code) is presented. The family of alternating knots is represented on the cubic lattice, that is, each alternating knot is composed of constant orthogonal straight-line segments and is described by means of a chain code. This chain code is represented by a numerical string of finite length over a finite alphabet, allowing the usage of formal-language techniques for alternating-knot representation. When an alternating knot is described by a chain, it is possible to obtain its mirroring image in an easy way. Also, we have a compression efficiency for representing alternating knots, because chain codes preserve information and allow a considerable data reduction.

A Characterization of Uniform Matroids

Brahim Chaourar — Thu, 01 Dec 2011 09:36:29 +0000

This paper gives a characterization of uniform matroids by means of locked subsets. Locked subsets are 2-connected subsets, their complements are 2-connected in the dual, and the minimum rank of both is 2. Locked subsets give the nontrivial facets of the bases polytope.

Irreducibility of the Tensor Product of Specializations of the Burau Representation of the Braid Groups

Mohammad N. Abdulrahim and Wiaam M. Zeid — Sun, 30 Oct 2011 09:32:56 +0000

The reduced Burau representation is a one-parameter representation of 𝐵𝑛, the braid group on 𝑛 strings. Specializing the parameter to nonzero complex number 𝑥 gives a representation 𝛽𝑛(𝑥): 𝐵𝑛→𝐺𝐿(ℂ𝑛−1), which is either irreducible or has an irreducible composition factor ̂‌𝛽𝑛(𝑥): 𝐵𝑛→𝐺𝐿(ℂ𝑛−2). In our paper, we let 𝑘≥2, and we determine a sufficient condition for the irreducibility of the tensor product of 𝑘 irreducible Burau representations. This is a generalization of our previous work concerning the cases 𝑘=2 and 𝑘=3.

Certain Transformation Formulae for Polybasic Hypergeometric Series

Pankaj Srivastava and Mohan Rudravarapu — Thu, 20 Oct 2011 11:54:56 +0000

Making use of Bailey's transformation and certain known summations of truncated series, an attempt has been made to establish transformation formulae involving polybasic hypergeometric series.

Finite Groups Whose Certain Subgroups of Prime Power Order Are 𝑆-Semipermutable

Mustafa Obaid — Tue, 11 Oct 2011 16:03:14 +0000

Let 𝐺 be a finite group. A subgroup 𝐻 of 𝐺 is said to be S-semipermutable in 𝐺 if 𝐻 permutes with every Sylow 𝑝-subgroup of 𝐺 with (𝑝,|𝐻|)=1. In this paper, we study the influence of S-permutability property of certain abelian subgroups of prime power order of a finite group on its structure.