ISRN Algebra http://www.hindawi.com The latest articles from Hindawi Publishing Corporation © 2014 , Hindawi Publishing Corporation . All rights reserved. A Note on Jordan Triple Higher *-Derivations on Semiprime Rings Wed, 09 Apr 2014 08:29:02 +0000 http://www.hindawi.com/journals/isrn.algebra/2014/365424/ We introduce the following notion. Let be the set of all nonnegative integers and let be a family of additive mappings of a *-ring R such that ; D is called a Jordan higher *-derivation (resp., a Jordan higher *-derivation) of R if (resp., ) for all and each . It is shown that the notions of Jordan higher *-derivations and Jordan triple higher *-derivations on a 6-torsion free semiprime *-ring are coincident. O. H. Ezzat Copyright © 2014 O. H. Ezzat. All rights reserved. When Is the Complement of the Comaximal Graph of a Commutative Ring Planar? Sun, 06 Apr 2014 09:40:03 +0000 http://www.hindawi.com/journals/isrn.algebra/2014/736043/ Let be a commutative ring with identity. In this paper we classify rings such that the complement of comaximal graph of is planar. We also consider the subgraph of the complement of comaximal graph of induced on the set of all nonunits of with the property that each element of is not in the Jacobson radical of and classify rings such that this subgraph is planar. S. Visweswaran and Jaydeep Parejiya Copyright © 2014 S. Visweswaran and Jaydeep Parejiya. All rights reserved. On 10-Centralizer Groups of Odd Order Tue, 01 Apr 2014 16:21:13 +0000 http://www.hindawi.com/journals/isrn.algebra/2014/607984/ Let be a group, and let denote the number of distinct centralizers of its elements. A group is called -centralizer if . In this paper, we investigate the structure of finite groups of odd order with and prove that there is no finite nonabelian group of odd order with . Z. Foruzanfar and Z. Mostaghim Copyright © 2014 Z. Foruzanfar and Z. Mostaghim. All rights reserved. An Investigation on Algebraic Structure of Soft Sets and Soft Filters over Residuated Lattices Thu, 13 Mar 2014 12:54:08 +0000 http://www.hindawi.com/journals/isrn.algebra/2014/635783/ We introduce the notion of soft filters in residuated lattices and investigate their basic properties. We investigate relations between soft residuated lattices and soft filter residuated lattices. The restricted and extended intersection (union), and -intersection, cartesian product, and restricted and extended difference of the family of soft filters residuated lattices are established. Also, we consider the set of all soft sets over a universe set and the set of parameters with respect to , (), and we study its structure. S. Rasouli and B. Davvaz Copyright © 2014 S. Rasouli and B. Davvaz. All rights reserved. Introduction to -Triple Systems Thu, 13 Mar 2014 08:17:45 +0000 http://www.hindawi.com/journals/isrn.algebra/2014/738154/ This paper introduces the category of -triple systems and studies some of their algebraic properties. Also provided is a functor from this category to the category of Leibniz algebras. Guy Roger Biyogmam Copyright © 2014 Guy Roger Biyogmam. All rights reserved. Divisibility Properties of the Fibonacci, Lucas, and Related Sequences Tue, 04 Mar 2014 13:28:53 +0000 http://www.hindawi.com/journals/isrn.algebra/2014/750325/ We use matrix techniques to give simple proofs of known divisibility properties of the Fibonacci, Lucas, generalized Lucas, and Gaussian Fibonacci numbers. Our derivations use the fact that products of diagonal matrices are diagonal together with Bezout’s identity. Thomas Jeffery and Rajesh Pereira Copyright © 2014 Thomas Jeffery and Rajesh Pereira. All rights reserved. On Generalized Jordan Triple -Higher Derivations in Prime Rings Wed, 22 Jan 2014 09:39:25 +0000 http://www.hindawi.com/journals/isrn.algebra/2014/684792/ Let be a ring and let be a Lie ideal of . Suppose that are endomorphisms of , and is the set of all nonnegative integers. A family of mappings is said to be a generalized -higher derivation (resp., generalized Jordan triple -higher derivation) of if there exists a -higher derivation of such that , the identity map on , , and (resp., hold for all and for every If the above conditions hold for all , then is said to be a generalized -higher derivation (resp., generalized Jordan triple -higher derivation) of into . In the present paper it is shown that if is a noncentral square closed Lie ideal of a prime ring of characteristic different from two, then every generalized Jordan triple -higher derivation of into is a generalized -higher derivation of into . Mohammad Ashraf and Almas Khan Copyright © 2014 Mohammad Ashraf and Almas Khan. All rights reserved. Generalized -Radical Supplemented Modules Tue, 14 Jan 2014 12:37:43 +0000 http://www.hindawi.com/journals/isrn.algebra/2014/603851/ Çalışıcı and Türkmen called a module generalized -supplemented if every submodule has a generalized supplement that is a direct summand of . Motivated by this, it is natural to introduce another notion that we called generalized -radical supplemented modules as a proper generalization of generalized -supplemented modules. In this paper, we obtain various properties of generalized -radical supplemented modules. We show that the class of generalized -radical supplemented modules is closed under finite direct sums. We attain that over a Dedekind domain a module is generalized -radical supplemented if and only if is generalized -radical supplemented. We completely determine the structure of these modules over left -rings. Moreover, we characterize semiperfect rings via generalized -radical supplemented modules. Burcu Nişancı Türkmen and Ali Pancar Copyright © 2014 Burcu Nişancı Türkmen and Ali Pancar. All rights reserved. Ioana's Superrigidity Theorem and Orbit Equivalence Relations Mon, 30 Dec 2013 12:01:50 +0000 http://www.hindawi.com/journals/isrn.algebra/2013/387540/ We give a survey of Adrian Ioana's cocycle superrigidity theorem for profinite actions of Property (T) groups and its applications to ergodic theory and set theory in this expository paper. In addition to a statement and proof of Ioana's theorem, this paper features the following: (i) an introduction to rigidity, including a crash course in Borel cocycles and a summary of some of the best-known superrigidity theorems; (ii) some easy applications of superrigidity, both to ergodic theory (orbit equivalence) and set theory (Borel reducibility); and (iii) a streamlined proof of Simon Thomas's theorem that the classification of torsion-free abelian groups of finite rank is intractable. Samuel Coskey Copyright © 2013 Samuel Coskey. All rights reserved. The EA-Dimension of a Commutative Ring Thu, 26 Dec 2013 18:27:04 +0000 http://www.hindawi.com/journals/isrn.algebra/2013/293207/ An elementary annihilator of a ring is an annihilator that has the form ; . We define the elementary annihilator dimension of the ring , denoted by , to be the upper bound of the set of all integers such that there is a chain of annihilators of . We use this dimension to characterize some zero-divisors graphs. Mosbah Eljeri Copyright © 2013 Mosbah Eljeri. All rights reserved. Some Theorems for Sigma Prime Rings with Differential Identities on Sigma Ideals Sun, 22 Dec 2013 13:05:18 +0000 http://www.hindawi.com/journals/isrn.algebra/2013/572690/ There has been considerable interest in the connection between the structure and the -structure of a ring, where denotes an involution on a ring. In this context, Oukhtite and Salhi (2006) introduced a new class or we can say an extension of prime rings in the form of -prime ring and proved several well-known theorems of prime rings for -prime rings. A continuous approach in the direction of -prime rings is still on. In this paper, we establish some results for -prime rings satisfying certain identities involving generalized derivations on -ideals. Finally, we give an example showing that the restrictions imposed on the hypothesis of the various theorems were not superfluous. Mohd Rais Khan and Mohd Mueenul Hasnain Copyright © 2013 Mohd Rais Khan and Mohd Mueenul Hasnain. All rights reserved. On Finite Nilpotent Matrix Groups over Integral Domains Mon, 16 Dec 2013 14:37:32 +0000 http://www.hindawi.com/journals/isrn.algebra/2013/638623/ We consider finite nilpotent groups of matrices over commutative rings. A general result concerning the diagonalization of matrix groups in the terms of simple conditions for matrix entries is proven. We also give some arithmetic applications for representations over Dedekind rings. Dmitry Malinin Copyright © 2013 Dmitry Malinin. All rights reserved. Simplicity and Commutative Bases of Derivations in Polynomial and Power Series Rings Thu, 12 Dec 2013 18:22:57 +0000 http://www.hindawi.com/journals/isrn.algebra/2013/560648/ The first part of the paper will describe a recent result of Retert in (2006) for and . This result states that if is a set of commute -derivations of such that both and the ring is -simple, then there is such that is -simple. As for applications, we obtain relationships with known results of A. Nowicki on commutative bases of derivations. Rene Baltazar Copyright © 2013 Rene Baltazar. All rights reserved. A Note on Solutions of Linear Systems Thu, 01 Aug 2013 10:16:05 +0000 http://www.hindawi.com/journals/isrn.algebra/2013/142124/ We will consider Rohde's general form of -inverse of a matrix . The necessary and sufficient condition for consistency of a linear system will be represented. We will also be concerned with the minimal number of free parameters in Penrose's formula for obtaining the general solution of the linear system. These results will be applied for finding the general solution of various homogenous and nonhomogenous linear systems as well as for different types of matrix equations. Branko Malešević, Ivana Jovović, Milica Makragić, and Biljana Radičić Copyright © 2013 Branko Malešević et al. All rights reserved. On Exact Sequence of Semimodules over Semirings Wed, 31 Jul 2013 08:51:01 +0000 http://www.hindawi.com/journals/isrn.algebra/2013/156485/ We introduce the notion of exact sequence of semimodules over semirings using maximal homomorphisms and generalize some results of module theory to semimodules over semirings. Indeed, we prove, “If is a split exact sequence of -semimodules and maximal -semimodule homomorphisms with is a strong subsemimodule of , then ”. Also, some results of commutative diagram of -semimodules and maximal -homomorphisms having exact rows are obtained. Jayprakash Ninu Chaudhari and Dipak Ravindra Bonde Copyright © 2013 Jayprakash Ninu Chaudhari and Dipak Ravindra Bonde. All rights reserved. Semiderivations Satisfying Certain Algebraic Identities on Jordan Ideals Tue, 18 Jun 2013 14:51:35 +0000 http://www.hindawi.com/journals/isrn.algebra/2013/738368/ Let be a ring. An additive mapping is called semiderivation of if there exists an endomorphism of such that and , for all in . Here we prove that if is a 2-torsion free -prime ring and a nonzero -Jordan ideal of such that for all , then either is commutative or for all . Moreover, we initiate the study of generalized semiderivations in prime rings. Vincenzo De Filippis, Abdellah Mamouni, and Lahcen Oukhtite Copyright © 2013 Vincenzo De Filippis et al. All rights reserved. On Hopf-Cyclic Cohomology and Cuntz Algebra Tue, 11 Jun 2013 15:25:28 +0000 http://www.hindawi.com/journals/isrn.algebra/2013/793637/ We demonstrate that Hopf cyclic cocycles, that is, cyclic cocycles with coefficients in stable anti-Yetter-Drinfeld modules, arise from invariant traces on certain ideals of Cuntz-type extension of the algebra. Andrzej Sitarz Copyright © 2013 Andrzej Sitarz. All rights reserved. A New Criterion for Affineness Wed, 20 Mar 2013 09:02:31 +0000 http://www.hindawi.com/journals/isrn.algebra/2013/786576/ 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. Jing Zhang Copyright © 2013 Jing Zhang. All rights reserved. On Cubic KU-Ideals of KU-Algebras Tue, 26 Feb 2013 16:36:32 +0000 http://www.hindawi.com/journals/isrn.algebra/2013/935905/ 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. Naveed Yaqoob, Samy M. Mostafa, and Moin A. Ansari Copyright © 2013 Naveed Yaqoob et al. All rights reserved. Idempotent Elements of the Endomorphism Semiring of a Finite Chain Tue, 26 Feb 2013 08:05:00 +0000 http://www.hindawi.com/journals/isrn.algebra/2013/120231/ 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. Ivan Trendafilov and Dimitrinka Vladeva Copyright © 2013 Ivan Trendafilov and Dimitrinka Vladeva. All rights reserved. Semientwining Structures and Their Applications Tue, 19 Feb 2013 08:58:38 +0000 http://www.hindawi.com/journals/isrn.algebra/2013/817919/ 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. Florin F. Nichita, Deepak Parashar, and Bartosz Zieliński Copyright © 2013 Florin F. Nichita et al. All rights reserved. A Generalization for -Cocycles Sun, 30 Dec 2012 08:57:07 +0000 http://www.hindawi.com/journals/isrn.algebra/2012/596741/ 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. Beishang Ren and Shixun Lin Copyright © 2012 Beishang Ren and Shixun Lin. All rights reserved. Commutativity Theorems for *-Prime Rings with Differential Identities on Jordan Ideals Sat, 29 Dec 2012 11:29:31 +0000 http://www.hindawi.com/journals/isrn.algebra/2012/729356/ 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. A. Mamouni, L. Oukhtite, and M. Samman Copyright © 2012 A. Mamouni et al. All rights reserved. Powers of Commutators and Anticommutators Mon, 24 Dec 2012 19:06:32 +0000 http://www.hindawi.com/journals/isrn.algebra/2012/302524/ 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. Stephen M. Buckley and Desmond MacHale Copyright © 2012 Stephen M. Buckley and Desmond MacHale. All rights reserved. On Generalized ()-Derivations in Semiprime Rings Mon, 03 Dec 2012 16:18:05 +0000 http://www.hindawi.com/journals/isrn.algebra/2012/120251/ 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 . Basudeb Dhara and Atanu Pattanayak Copyright © 2012 Basudeb Dhara and Atanu Pattanayak. All rights reserved. Growth for Algebras Satisfying Polynomial Identities Wed, 21 Nov 2012 14:05:59 +0000 http://www.hindawi.com/journals/isrn.algebra/2012/170697/ 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. Amitai Regev Copyright © 2012 Amitai Regev. All rights reserved. On Pre-Hilbert Noncommutative Jordan Algebras Satisfying β€–π‘₯2β€–=β€–π‘₯β€–2 Thu, 23 Aug 2012 10:02:53 +0000 http://www.hindawi.com/journals/isrn.algebra/2012/328752/ Let 𝐴 be a real or complex algebra. Assuming that a vector space 𝐴 is endowed with a pre-Hilbert norm β€–β‹…β€– satisfying β€–π‘₯2β€–=β€–π‘₯β€–2 for all π‘₯∈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,β„‚,ℍ, or 𝕆, and 𝐴 is isomorphic to β„‚ in the last three cases. These last cases permit us to show that if 𝐴 is a complex pre-Hilbert noncommutative Jordan algebra satisfying β€–π‘₯2β€–=β€–π‘₯β€–2 for all π‘₯∈𝐴, then 𝐴 is finite dimensional and is isomorphic to β„‚. Moreover, we give an example of an infinite-dimensional real pre-Hilbert Jordan algebra with divisors of zero and satisfying β€–π‘₯2β€–=β€–π‘₯β€–2 for all π‘₯∈𝐴. Mohamed Benslimane and Abdelhadi Moutassim Copyright © 2012 Mohamed Benslimane and Abdelhadi Moutassim. All rights reserved. A Gelfand Model for Weyl Groups of Type 𝐷2𝑛 Tue, 07 Aug 2012 13:35:09 +0000 http://www.hindawi.com/journals/isrn.algebra/2012/658201/ 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(β„‚), where β„‚ is the field of complex numbers, it has been proved that each 𝐺-module over β„‚ is isomorphic to a 𝐺-submodule in the polynomial ring β„‚[π‘₯1,…,π‘₯𝑛], and taking the space of zeros of certain 𝐺-invariant operators in the Weyl algebra, a finite-dimensional 𝐺-space 𝒩𝐺 in β„‚[π‘₯1,…,π‘₯𝑛] can be obtained, which contains all the simple 𝐺-modules over β„‚. 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. José O. Araujo, Luis C. Maiarú, and Mauro Natale Copyright © 2012 José O. Araujo et al. All rights reserved. Cogredient Standard Forms of Symmetric Matrices over Galois Rings of Odd Characteristic Thu, 19 Jul 2012 17:52:41 +0000 http://www.hindawi.com/journals/isrn.algebra/2012/520148/ Let 𝑅=GR(𝑝𝑠,π‘π‘ π‘š) be a Galois ring of characteristic 𝑝𝑠 and cardinality π‘π‘ π‘š, where 𝑠 and π‘š 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. Yonglin Cao Copyright © 2012 Yonglin Cao. All rights reserved. On the Completely Positive and Positive Semidefinite-Preserving Cones—Part III Mon, 02 Jul 2012 14:20:00 +0000 http://www.hindawi.com/journals/isrn.algebra/2012/704960/ This paper studies the extremals and other faces of the completely positive and positive semidefinite-preserving linear transformations. Andrew J. Klimas and Richard D. Hill Copyright © 2012 Andrew J. Klimas and Richard D. Hill. All rights reserved.