Algebra http://www.hindawi.com The latest articles from Hindawi Publishing Corporation © 2014 , Hindawi Publishing Corporation . All rights reserved. On the Torsion Units of Integral Adjacency Algebras of Finite Association Schemes Tue, 16 Dec 2014 07:06:24 +0000 http://www.hindawi.com/journals/algebra/2014/842378/ Torsion units of group rings have been studied extensively since the 1960s. As association schemes are generalization of groups, it is natural to ask about torsion units of association scheme rings. In this paper we establish some results about torsion units of association scheme rings analogous to basic results for torsion units of group rings. Allen Herman and Gurmail Singh Copyright © 2014 Allen Herman and Gurmail Singh. All rights reserved. Zero Divisor Graph for the Ring of Eisenstein Integers Modulo Mon, 15 Dec 2014 07:56:40 +0000 http://www.hindawi.com/journals/algebra/2014/146873/ Let be the ring of Eisenstein integers modulo . In this paper we study the zero divisor graph . We find the diameters and girths for such zero divisor graphs and characterize for which the graph is complete, complete bipartite, bipartite, regular, Eulerian, Hamiltonian, or chordal. Osama Alkam and Emad Abu Osba Copyright © 2014 Osama Alkam and Emad Abu Osba. All rights reserved. Right -Weakly Regular -Semirings Mon, 15 Dec 2014 00:10:04 +0000 http://www.hindawi.com/journals/algebra/2014/948032/ The concepts of a -idempotent -semiring, a right -weakly regular -semiring, and a right pure -ideal of a -semiring are introduced. Several characterizations of them are furnished. R. D. Jagatap Copyright © 2014 R. D. Jagatap. All rights reserved. Injectivity of the Composition Operators of Étale Mappings Wed, 10 Dec 2014 08:58:23 +0000 http://www.hindawi.com/journals/algebra/2014/782973/ Let be a topological space. The semigroup of all the étale mappings of (the local homeomorphisms ) is denoted by . If , then the -right (left) composition operator on is defined by   , . When are the composition operators injective? The Problem originated in a new approach to study étale polynomial mappings and in particular the two-dimensional Jacobian conjecture. This approach constructs a fractal structure on the semigroup of the (normalized) Keller mappings and outlines a new method of a possible attack on this open problem (in preparation). The construction uses the left composition operator and the injectivity problem is essential. In this paper we will completely solve the injectivity problems of the two composition operators for (normalized) Keller mappings. We will also solve the much easier surjectivity problem of these composition operators. Ronen Peretz Copyright © 2014 Ronen Peretz. All rights reserved. Finitely Generated Modules over Group Rings of a Direct Product of Two Cyclic Groups Mon, 01 Dec 2014 11:39:47 +0000 http://www.hindawi.com/journals/algebra/2014/256020/ Let be a commutative field of characteristic and let , where and are two finite cyclic groups. We give some structure results of finitely generated -modules in the case where the order of is divisible by . Extensions of modules are also investigated. Based on these extensions and in the same previous case, we show that -modules satisfying some conditions have a fairly simple form. Ahmed Najim and Mohammed Elhassani Charkani Copyright © 2014 Ahmed Najim and Mohammed Elhassani Charkani. All rights reserved. The Hilbert-Kunz Function for Binomial Hypersurfaces Thu, 27 Nov 2014 07:58:19 +0000 http://www.hindawi.com/journals/algebra/2014/525467/ I give an iterative closed form formula for the Hilbert-Kunz function for any binomial hypersurface in general, over any field of arbitrary positive characteristic. I prove that the Hilbert-Kunz multiplicity associated with any binomial hypersurface over any field of arbitrary positive characteristic is rational. As an example, I also prove the well known fact that for 1-dimensional binomial hypersurfaces the Hilbert-Kunz multiplicity is a positive integer and give a precise account of the integer. Shyamashree Upadhyay Copyright © 2014 Shyamashree Upadhyay. All rights reserved. Residuation Properties and Weakly Primary Elements in Lattice Modules Thu, 27 Nov 2014 06:32:19 +0000 http://www.hindawi.com/journals/algebra/2014/858323/ We obtain some elementary residuation properties in lattice modules and obtain a relation between a weakly primary element in a lattice module and weakly prime element of a multiplicative lattice . C. S. Manjarekar and U. N. Kandale Copyright © 2014 C. S. Manjarekar and U. N. Kandale. All rights reserved. Jordan Higher Derivable Mappings on Rings Wed, 19 Nov 2014 00:00:00 +0000 http://www.hindawi.com/journals/algebra/2014/672387/ Let be a ring. We say that a family of maps is a Jordan higher derivable map (without assumption of additivity) on if (the identity map on ) and hold for all and for each . In this paper, we show that every Jordan higher derivable map on a ring under certain assumptions becomes a higher derivation. As its application, we get that every Jordan higher derivable map on Banach algebra is an additive higher derivation. Mohammad Ashraf and Nazia Parveen Copyright © 2014 Mohammad Ashraf and Nazia Parveen. All rights reserved. Some Properties of Multiplicative -Rings of Polynomials over Multiplicative Hyperrings Mon, 27 Oct 2014 13:16:50 +0000 http://www.hindawi.com/journals/algebra/2014/392902/ The set of all polynomials , over a multiplicative hyperring , form a commutative group with respect to the component-wise addition (+) of the polynomials. For polynomials in , is a set of polynomials whose th components are chosen from the set , where and are the th and the th components of and , respectively. A multiplicative hyperring is polynomially structured if the hyperstructure is a multiplicative -ring. The purpose of the paper is to study the properties of the multiplicative -ring , corresponding to those of a polynomially structured multiplicative hyperring . Utpal Dasgupta Copyright © 2014 Utpal Dasgupta. All rights reserved. -Prime and -Primary Elements in Multiplicative Lattices Thu, 09 Oct 2014 09:48:11 +0000 http://www.hindawi.com/journals/algebra/2014/890312/ We investigate -prime and -primary elements in a compactly generated multiplicative lattice . By a counterexample, it is shown that a -primary element in need not be primary. Some characterizations of -primary and -prime elements in are obtained. Finally, some results for almost prime and almost primary elements in with characterizations are obtained. C. S. Manjarekar and A. V. Bingi Copyright © 2014 C. S. Manjarekar and A. V. Bingi. All rights reserved. The Relatively Free Groups Satisfy Noncentral Commutative Transitivity Wed, 01 Oct 2014 09:37:35 +0000 http://www.hindawi.com/journals/algebra/2014/379030/ We prove that a free group, , relative to the variety, , of all groups simultaneously nilpotent of class at most and metabelian is such that the centralizer of every noncentral element is abelian. We relate that result to the model theory of such groups as well as a quest to find a relative analog in of a classical theorem of Benjamin Baumslag. We also touch briefly on similar considerations in the varieties of nilpotent groups. Anthony M. Gaglione, Seymour Lipschutz, and Dennis Spellman Copyright © 2014 Anthony M. Gaglione et al. All rights reserved. The Matrix Equation over Fields or Rings Wed, 01 Oct 2014 06:34:40 +0000 http://www.hindawi.com/journals/algebra/2014/745029/ Let and let be an algebraically closed field with characteristic 0 or greater than . We show that if and satisfy , then are simultaneously triangularizable. Let be a reduced ring such that is not a zero divisor and let be a generic matrix over ; we show that is the sole solution of . Let be a commutative ring with unity; let be similar to such that, for every is not a zero divisor. If is a nilpotent solution of where , then . Gerald Bourgeois Copyright © 2014 Gerald Bourgeois. All rights reserved. On Almost Semiprime Submodules Wed, 10 Sep 2014 11:36:09 +0000 http://www.hindawi.com/journals/algebra/2014/752858/ We introduce the concept of almost semiprime submodules of unitary modules over a commutative ring with nonzero identity. We investigate some basic properties of almost semiprime and weakly semiprime submodules and give some characterizations of them, especially for (finitely generated faithful) multiplication modules. Farkhonde Farzalipour Copyright © 2014 Farkhonde Farzalipour. All rights reserved. The Reducibility of a Special Binary Pentanomial Tue, 02 Sep 2014 12:43:02 +0000 http://www.hindawi.com/journals/algebra/2014/482837/ Swan’s theorem determines the parity of the number of irreducible factors of a binary trinomial. In this work, we study the parity of the number of irreducible factors for a special binary pentanomial with even degree , where , and exactly one of  ,  and   is odd. This kind of irreducible pentanomials can be used for a fast implementation of trace and square root computations in finite fields of characteristic 2. Ryul Kim and Yun Mi Kim Copyright © 2014 Ryul Kim and Yun Mi Kim. All rights reserved. Unions of Parafree Lie Algebras Wed, 13 Aug 2014 09:18:21 +0000 http://www.hindawi.com/journals/algebra/2014/385397/ We consider unions of parafree Lie algebras and we prove that such unions are again parafree under some conditions. Naime Ekici and Zehra Velioğlu Copyright © 2014 Naime Ekici and Zehra Velioğlu. All rights reserved. A Study of Inverse Problems Based on Two Kinds of Special Matrix Equations in Euclidean Space Mon, 26 May 2014 00:00:00 +0000 http://www.hindawi.com/journals/algebra/2014/392467/ Two special classes of symmetric coefficient matrices were defined based on characteristics matrix; meanwhile, the expressions of the solution to inverse problems are given and the conditions for the solvability of these problems are studied relying on researching. Finally, the optimal approximation solution of these problems is provided. Rui Huang, Xiaodong Wu, Ruihe Wang, and Hui Li Copyright © 2014 Rui Huang et al. All rights reserved. Demazure Descent and Representations of Reductive Groups Sun, 25 May 2014 10:58:34 +0000 http://www.hindawi.com/journals/algebra/2014/823467/ We introduce the notion of Demazure descent data on a triangulated category and define the descent category for such data. We illustrate the definition by our basic example. Let be a reductive algebraic group with a Borel subgroup . Demazure functors form Demazure descent data on and the descent category is equivalent to . Sergey Arkhipov and Tina Kanstrup Copyright © 2014 Sergey Arkhipov and Tina Kanstrup. All rights reserved. On Determinantal Varieties of Hankel Matrices Mon, 28 Apr 2014 08:00:28 +0000 http://www.hindawi.com/journals/algebra/2014/970157/ Let be a class of Hankel matrices whose entries, depending on a given matrix , are linear forms in variables with coefficients in a finite field . For every matrix in , it is shown that the varieties specified by the leading minors of orders from 1 to have the same number of points in . Further properties are derived, which show that sets of varieties, tied to a given Hankel matrix, resemble a set of hyperplanes as regards the number of points of their intersections. Edoardo Ballico and Michele Elia Copyright © 2014 Edoardo Ballico and Michele Elia. All rights reserved. Simplices in the Endomorphism Semiring of a Finite Chain Sun, 27 Apr 2014 08:50:15 +0000 http://www.hindawi.com/journals/algebra/2014/263605/ We establish new results concerning endomorphisms of a finite chain, if the cardinality of the image of such endomorphism is no more than some fixed number. The semiring of all such endomorphisms can be seen as a simplex whose vertices are the constant endomorphisms. We explore the properties of these simplices. Ivan Trendafilov Copyright © 2014 Ivan Trendafilov. All rights reserved. On Integral Manifolds for Leibniz Algebras Thu, 24 Apr 2014 08:37:41 +0000 http://www.hindawi.com/journals/algebra/2014/875981/ We discuss several partial solutions to the so-called “coquecigrue problem” of Loday; these solutions parallel, but also generalize in several directions, the classical Lie group-Lie algebra correspondence. Our study highlights some clear similarities between the split and nonsplit cases and leads us to a general unifying scheme that provides an answer to the problem of the algebraic structure of a coquecigrue. Juan Monterde and Fausto Ongay Copyright © 2014 Juan Monterde and Fausto Ongay. All rights reserved. A Construction of Bent Functions of Variables from a Bent Function of Variables and Its Cyclic Shifts Thu, 17 Apr 2014 10:08:48 +0000 http://www.hindawi.com/journals/algebra/2014/701298/ We present a method to iteratively construct new bent functions of variables from a bent function of variables and its cyclic shift permutations using minterms of variables and minterms of 2 variables. In addition, we provide the number of bent functions of variables that we can obtain by applying the method here presented, and finally we compare this method with a previous one introduced by us in 2008 and with the Rothaus and Maiorana-McFarland constructions. Joan-Josep Climent, Francisco J. García, and Verónica Requena Copyright © 2014 Joan-Josep Climent et al. All rights reserved. Vertex Coalgebras, Coassociator, and Cocommutator Formulas Sun, 02 Mar 2014 13:26:01 +0000 http://www.hindawi.com/journals/algebra/2014/861768/ Based on the definition of vertex coalgebra introduced by Hubbard, 2009, we prove that this notion can be reformulated using coskew symmetry, coassociator and cocommutator formulas without restrictions on the grading. We also prove that a vertex coalgebra can be defined in terms of dual versions of the axioms of Lie conformal algebra and differential algebra. Florencia Orosz Hunziker and José I. Liberati Copyright © 2014 Florencia Orosz Hunziker and José I. Liberati. All rights reserved. On Ordered Quasi-Gamma-Ideals of Regular Ordered Gamma-Semigroups Mon, 09 Dec 2013 16:00:41 +0000 http://www.hindawi.com/journals/algebra/2013/565848/ We introduce the notion of ordered quasi--ideals of regular ordered -semigroups and study the basic properties of ordered quasi--ideals of ordered -semigroups. We also characterize regular ordered -semigroups in terms of their ordered quasi--ideals, ordered right -ideals, and left -ideals. Finally, we have shown that (i) a partially ordered -semigroup is regular if and only if for every ordered bi--ideal , every ordered -ideal , and every ordered quasi--ideal , we have and (ii) a partially ordered -semigroup is regular if and only if for every ordered quasi--ideal , every ordered left -ideal , and every ordered right--ideal , we have that . M. Y. Abbasi and Abul Basar Copyright © 2013 M. Y. Abbasi and Abul Basar. All rights reserved. Commutative and Bounded BE-algebras Thu, 05 Dec 2013 14:33:53 +0000 http://www.hindawi.com/journals/algebra/2013/473714/ We introduce the notions of the commutative and bounded BE-Algebras. We give some related properties of them. Zekiye Çiloğlu and Yılmaz Çeven Copyright © 2013 Zekiye Çiloğlu and Yılmaz Çeven. All rights reserved. On the Jacobson Radical of an -Semiring Mon, 07 Oct 2013 14:15:37 +0000 http://www.hindawi.com/journals/algebra/2013/272104/ The notion of -ary semimodules is introduced so that the Jacobson radical of an -semiring is studied and some well-known results concerning the Jacobson radical of a ring (a semiring or a ternary semiring) are generalized to an -semiring. Yongwen Zhu Copyright © 2013 Yongwen Zhu. All rights reserved. A Study on Fuzzy Ideals of -Groups Thu, 26 Sep 2013 11:15:48 +0000 http://www.hindawi.com/journals/algebra/2013/594636/ Using the idea of the new sort of fuzzy subnear-ring of a near-ring, fuzzy subgroups, and their generalizations defined by various researchers, we try to introduce the notion of ()-fuzzy ideals of -groups. These fuzzy ideals are characterized by their level ideals, and some other related properties are investigated. B. Davvaz and O. Ratnabala Devi Copyright © 2013 B. Davvaz and O. Ratnabala Devi. All rights reserved. Two Interacting Coordinate Hopf Algebras of Affine Groups of Formal Series on a Category Thu, 19 Sep 2013 13:14:34 +0000 http://www.hindawi.com/journals/algebra/2013/370618/ A locally finite category is defined as a category in which every arrow admits only finitely many different ways to be factorized by composable arrows. The large algebra of such categories over some fields may be defined, and with it a group of invertible series (under multiplication). For certain particular locally finite categories, a substitution operation, generalizing the usual substitution of formal power series, may be defined, and with it a group of reversible series (invertible under substitution). Moreover, both groups are actually affine groups. In this contribution, we introduce their coordinate Hopf algebras which are both free as commutative algebras. The semidirect product structure obtained from the action of reversible series on invertible series by anti-automorphisms gives rise to an interaction at the level of their coordinate Hopf algebras under the form of a smash coproduct. Laurent Poinsot Copyright © 2013 Laurent Poinsot. All rights reserved. Construction and Composition of Rooted Trees via Descent Functions Tue, 27 Aug 2013 16:06:44 +0000 http://www.hindawi.com/journals/algebra/2013/543913/ We propose a novel approach for studying rooted trees by using functions that we will call descent functions. We provide a construction method for rooted trees that allows to study their properties through the use of descent functions. Moreover, in this way, we are able to compose rooted trees with each other. Such a new composition of rooted trees is a very powerful tool applied in this paper in order to obtain important results as the creation of new rational and Pythagorean trees. Marco Abrate, Stefano Barbero, Umberto Cerruti, and Nadir Murru Copyright © 2013 Marco Abrate et al. All rights reserved. Some Results on Strict Graded Categorical Groups Mon, 26 Aug 2013 09:25:24 +0000 http://www.hindawi.com/journals/algebra/2013/306978/ We present some applications of strict graded categorical groups to the construction of the obstruction of an equivariant kernel and to the classification of equivariant group extensions which are central ones. The composition of a graded categorical group and an equivariant group homomorphism is also determined. Nguyen Tien Quang and Che Thi Kim Phung Copyright © 2013 Nguyen Tien Quang and Che Thi Kim Phung. All rights reserved. A Characterization of Projective Special Unitary Group U3(7) by nse Sat, 24 Aug 2013 08:07:54 +0000 http://www.hindawi.com/journals/algebra/2013/983186/ Let a group and be the set of element orders of . Let and let be the number of elements of order in . Let nse. In Khatami et al. and Liu's works, and are uniquely determined by nse. In this paper, we prove that if is a group such that nse = nse, then . Shitian Liu Copyright © 2013 Shitian Liu. All rights reserved.