ISRN Algebra
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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 6torsion 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 10Centralizer 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 bestknown 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 torsionfree abelian groups of finite rank is intractable.
Samuel Coskey
Copyright © 2013 Samuel Coskey. All rights reserved.

The EADimension 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 zerodivisors 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 wellknown 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 2torsion 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 HopfCyclic 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 antiYetterDrinfeld modules, arise from invariant traces on certain ideals of Cuntztype 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 KUIdeals of KUAlgebras
Tue, 26 Feb 2013 16:36:32 +0000
http://www.hindawi.com/journals/isrn.algebra/2013/935905/
We introduce the notion of cubic KUideals of KUalgebras and several results are presented in this regard. The image, preimage, and cartesian product of cubic KUideals of KUalgebras 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 wellknown 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 YangBaxter 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 quasibialgebra and also show properties of type II cocycles and some results about weak quasibialgebras, for instance, construct a new structure of tensor product algebra over a module algebra on weak quasibialgebras.
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 PreHilbert Noncommutative Jordan Algebras Satisfying βπ₯2β=βπ₯β2
Thu, 23 Aug 2012 10:02:53 +0000
http://www.hindawi.com/journals/isrn.algebra/2012/328752/
Let π΄ be a real or complex algebra. Assuming that a vector space π΄ is endowed with a preHilbert norm ββ
β satisfying βπ₯2β=βπ₯β2 for all π₯βπ΄. We prove that π΄ is finite dimensional in the following cases. (1) π΄ is a real weakly alternative algebra without divisors of zero. (2) π΄ is a complex powers associative algebra. (3) π΄ is a complex flexible algebraic algebra. (4) π΄ is a complex Jordan algebra. In the first case π΄ is isomorphic to β,β,β, or π, and π΄ is isomorphic to β in the last three cases. These last cases permit us to show that if π΄ is a complex preHilbert noncommutative Jordan algebra satisfying βπ₯2β=βπ₯β2 for all π₯βπ΄, then π΄ is finite dimensional and is isomorphic to β. Moreover, we give an example of an infinitedimensional real preHilbert Jordan algebra with divisors of zero and satisfying βπ₯2β=βπ₯β2 for all π₯βπ΄.
Mohamed Benslimane and Abdelhadi Moutassim
Copyright © 2012 Mohamed Benslimane and Abdelhadi Moutassim. All rights reserved.

A Gelfand Model for Weyl Groups of Type π·2π
Tue, 07 Aug 2012 13:35:09 +0000
http://www.hindawi.com/journals/isrn.algebra/2012/658201/
A Gelfand model for a finite group πΊ is a complex representation of πΊ, which is isomorphic to the direct sum of all irreducible representations of πΊ. When πΊ is isomorphic to a subgroup of πΊπΏπ(β), where β is the field of complex numbers, it has been proved that each πΊmodule over β is isomorphic to a πΊsubmodule in the polynomial ring β[π₯1,β¦,π₯π], and taking the space of zeros of certain πΊinvariant operators in the Weyl algebra, a finitedimensional πΊspace π©πΊ in β[π₯1,β¦,π₯π] can be obtained, which contains all the simple πΊmodules over β. This type of representation has been named polynomial model. It has been proved that when πΊ is a Coxeter group, the polynomial model is a Gelfand model for πΊ if, and only if, πΊ has not an irreducible factor of type π·2π, πΈ7, or πΈ8. This paper presents a model of Gelfand for a Weyl group of type π·2π 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 π
=GR(ππ ,ππ π) be a Galois ring of characteristic ππ and cardinality ππ π, where π
and π
are positive integers and π
is an odd prime number. Two kinds of cogredient standard forms of symmetric matrices over π
are given, and an explicit formula to count the number of all distinct cogredient classes of symmetric matrices over π
is obtained.
Yonglin Cao
Copyright © 2012 Yonglin Cao. All rights reserved.

On the Completely Positive and Positive SemidefinitePreserving 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 semidefinitepreserving linear transformations.
Andrew J. Klimas and Richard D. Hill
Copyright © 2012 Andrew J. Klimas and Richard D. Hill. All rights reserved.