Algebra The latest articles from Hindawi Publishing Corporation © 2014 , Hindawi Publishing Corporation . All rights reserved. -Prime and -Primary Elements in Multiplicative Lattices Thu, 09 Oct 2014 09:48:11 +0000 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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. On Quasi--Dense Submodules and -Pure Envelopes of QTAG Modules Mon, 29 Jul 2013 10:40:07 +0000 A module over an associative ring with unity is a QTAG module if every finitely generated submodule of any homomorphic image of is a direct sum of uniserial modules. There are many fascinating properties of QTAG modules of which -pure submodules and high submodules are significant. A submodule is quasi--dense in if is -divisible, for every -pure submodule of containing Here we study these submodules and obtain some interesting results. Motivated by -neat envelope, we also define -pure envelope of a submodule as the -pure submodule if has no direct summand containing We find that -pure envelopes of have isomorphic basic submodules, and if is the direct sum of uniserial modules, then all -pure envelopes of are isomorphic. Alveera Mehdi, Fahad Sikander, and Firdhousi Begum Copyright © 2013 Alveera Mehdi et al. All rights reserved. A Note on Hobby’s Theorem of Finite Groups Sun, 07 Jul 2013 08:56:25 +0000 It is well known that the Frattini subgroups of any finite groups are nilpotent. If a finite group is not nilpotent, it is not the Frattini subgroup of a finite group. In this paper, we mainly discuss what kind of finite nilpotent groups cannot be the Frattini subgroup of some finite groups and give some results. Moreover, we generalize Hobby’s Theorem. Qingjun Kong Copyright © 2013 Qingjun Kong. All rights reserved. The Depth of Generalized Full Terms and Generalized Full Hypersubstitutions Wed, 26 Jun 2013 15:17:15 +0000 We generalize the concept of a full term which was introduced by Denecke and Changphas and the concept of the depth of a hypersubstitution which was introduced by Denecke et al. to the depth of a generalized full term and the depth of a generalized full hypersubstitution and then derive the formular for its depth. Sarawut Phuapong and Sorasak Leeratanavalee Copyright © 2013 Sarawut Phuapong and Sorasak Leeratanavalee. All rights reserved. Bound on the Minimum Eigenvalue of -Matrices Involving Hadamard Products Wed, 19 Jun 2013 08:49:09 +0000 We present a new lower bound on the minimum eigenvalue of -matrices involving Hadamard products , and we show that our lower bound is larger than the lower bound . Three examples verify our result. Kun Du, Guiding Gu, and Guo Liu Copyright © 2013 Kun Du et al. All rights reserved. Finite Groups with Some -Supplemented Subgroups Tue, 18 Jun 2013 14:23:16 +0000 Let be a subgroup of a finite group , a prime dividing the order of , and a Sylow -subgroup of for prime We say that is -supplemented in if there is a subgroup of such that and where denotes the subgroup of generated by all those subgroups of which are -quasinormally embedded in In this paper, we characterize -nilpotency and supersolvability of under the assumption that all maximal subgroups of are -supplemented in . Guo Zhong, Liying Yang, Huaquan Wei, Xuanlong Ma, and Jiayi Xia Copyright © 2013 Guo Zhong et al. All rights reserved. On -Cogenerated Commutative Unital -Algebras Wed, 05 Jun 2013 13:38:42 +0000 Gelfand-Naimark's theorem states that every commutative -algebra is isomorphic to a complex valued algebra of continuous functions over a suitable compact space. We observe that for a completely regular space , is dense--separable if and only if is -cogenerated if and only if every family of maximal ideals of with zero intersection has a subfamily with cardinal number less than and zero intersection. This gives a simple characterization of -cogenerated commutative unital -algebras via their maximal ideals. Ehsan Momtahan Copyright © 2013 Ehsan Momtahan. All rights reserved. Wronskian Envelope of a Lie Algebra Wed, 29 May 2013 10:56:18 +0000 The famous Poincaré-Birkhoff-Witt theorem states that a Lie algebra, free as a module, embeds into its associative envelope—its universal enveloping algebra—as a sub-Lie algebra for the usual commutator Lie bracket. However, there is another functorial way—less known—to associate a Lie algebra to an associative algebra and inversely. Any commutative algebra equipped with a derivation , that is, a commutative differential algebra, admits a Wronskian bracket under which it becomes a Lie algebra. Conversely, to any Lie algebra a commutative differential algebra is universally associated, its Wronskian envelope, in a way similar to the associative envelope. This contribution is the beginning of an investigation of these relations between Lie algebras and differential algebras which is parallel to the classical theory. In particular, we give a sufficient condition under which a Lie algebra may be embedded into its Wronskian envelope, and we present the construction of the free Lie algebra with this property. Laurent Poinsot Copyright © 2013 Laurent Poinsot. All rights reserved. Recent Progress on Submersions: A Survey and New Properties Sun, 12 May 2013 13:54:14 +0000 This paper is a survey about recent progress on submersive morphisms of schemes combined with new results that we prove. They concern the class of quasicompact universally subtrusive morphisms that we introduced about 30 years ago. They are revisited in a recent paper by Rydh, with substantial complements and key results. We use them to show Artin-Tate-like results about the 14th problem of Hilbert, for a base scheme either Noetherian or the spectrum of a valuation domain. We look at faithfully flat morphisms and get “almost” Artin-Tate-like results by considering the Goldman (finite type) points of a scheme. Bjorn Poonen recently proved that universally closed morphisms are quasicompact. By introducing incomparable morphisms of schemes, we are able to characterize universally closed surjective morphisms that are either integral or finite. Next we consider pure morphisms of schemes introduced by Mesablishvili. In the quasicompact case, they are universally schematically dominant morphisms. This leads us to a characterization of universally subtrusive morphisms by purity. Some results on the schematically dominant property are given. The paper ends with properties of monomorphisms and topological immersions, a dual notion of submersions. Gabriel Picavet Copyright © 2013 Gabriel Picavet. All rights reserved. The Cyclic Graph of a Finite Group Tue, 16 Apr 2013 08:14:47 +0000 The cyclic graph of a finite group is as follows: take as the vertices of and join two distinct vertices and if is cyclic. In this paper, we investigate how the graph theoretical properties of affect the group theoretical properties of . First, we consider some properties of and characterize certain finite groups whose cyclic graphs have some properties. Then, we present some properties of the cyclic graphs of the dihedral groups and the generalized quaternion groups for some . Finally, we present some parameters about the cyclic graphs of finite noncyclic groups of order up to . Xuan Long Ma, Hua Quan Wei, and Guo Zhong Copyright © 2013 Xuan Long Ma et al. All rights reserved.