Classification of a subclass of low-dimensional complex filiform Leibniz algebras.

We give a complete classification of a subclass of complex filiform Leibniz algebras obtained from naturally graded non-Lie filiform Leibniz algebras. The isomorphism criteria in terms of invariant functions are given

On hilbert property of rings.

In this paper, we study a Hilbert property with respect to the skew polynomial extensions. A property ∗ of a ring R is said to be the Hilbert property if its polynomial extension possesses the same property ∗

On isomorphism classes of a subclass of filiform Leibniz algebras in dimension 10

Some classes of filiform Leibniz algebras arising from the naturally graded non Lie filiform Leibniz algebras were introduced by Ayupov and Omirov in 2006. One of them is the first class of filiform Leibniz algebras and has been denoted by FLBn in fixed dimension n.Rakhimov and Bekbaev have suggested an approach of classifying FLBn based on algebraic invariants method. In terms of classification, it has been classified up to dimension 9 and the main purpose of this paper is to use method of invariants and apply to dimension 10. To complete this goal, we set up an isomorphism criterion, here we introduced some new functions to avoid big expressions involved in isomorphism criterion. From the criterion some lists of disjoint subsets from FLB10 are obtained. These disjoint subsets are represented as a union of parametric family of orbits or just single orbits. For parametric cases, the invariants will be given

On Prime-Gamma-Near-Rings with Generalized Derivations

Let be a 2-torsion free prime Γ-near-ring with center (). Let (,) and (,ℎ) be two generalized derivations on . We prove the following results: (i) if ([,])=0 or ([,])=±[,] or 2()∈() for all ,∈, ∈Γ, then is a commutative Γ-ring. (ii) If ∈ and [(),]=0 for all ∈, ∈Γ, then ()∈(). (iii) If (,ℎ) acts as a generalized derivation on , then =0 or =0

A new mathematical evaluation of smoking problem based on an algebraic statistical method

Smoking problem is considered as one of the hot topics for many years. In spite of overpowering facts about the dangers, smoking is still a bad habit widely spread and socially accepted. Many people start smoking during their gymnasium period. The discovery of the dangers of smoking gave a warning sign of danger for individuals. There are different statistical methods used to analyze the dangers of smoking. In this study, we apply an algebraic statistical method to analyze and classify real data using Markov basis for the independent model on the contingency table. Results show that the Markov basis based classification is able to distinguish different date elements. Moreover, we check our proposed method via information theory by utilizing the Shannon formula to illustrate which one of these alternative tables is the best in term of independent

On central extensions of associative dialgebras

The concept of central extensions plays an important in constructing extensions of algebras. This technique has been successfully used in the classification problem of certain classes of algebras. In 1978 Skjelbred and Sund reduced the classification of nilpotent Lie algebras in a given dimension to the study of orbits under the action of automorphism group on the space of second degree cohomology of a smaller Lie algebra with coefficients in a trivial module. Then W. de Graaf applied the Skjelbred and Sund method to the classification problem of low-dimensional nilpotent Lie and associative algebras over some fields. The main purpose of this note is to establish elementary properties of central extensions of associative dialgebras and apply the above mentioned method to the classification of low dimensional nilpotent associative dialgebras

Generalized Derivations in Semiprime Gamma Rings

Let M be a 2-torsion-free semiprime Γ-ring satisfying the condition aαbβc=aβbαc for all a,b,c∈M,  α,β∈Γ, and let D:M→M be an additive mapping such that D(xαx)=D(x)αx+xαd(x) for all x∈M,  α∈Γ and for some derivation d of M. We prove that D is a generalized derivation