In this paper, we proved that if R is a prime ring, U be a nonzero Lie ideal of R , d be a nonzero (?,?)-derivation of R. Then if Ua?Z(R) (or aU?Z(R)) for a?R, then either or U is commutative Also, we assumed that Uis a ring to prove that: (i) If Ua?Z(R) (or aU?Z(R)) for a?R, then either a=0 or U is commutative. (ii) If ad(U)=0 (or d(U)a=0) for a?R, then either a=0 or U is commutative. (iii) If d is a homomorphism on U such that ad(U) ?Z(R)(or d(U)a?Z(R), then a=0 or U is commutative.

     The new type of paranormal operators that have been defined in this study on the Hilbert space, is  paranormal operators. In this paper we introduce and discuss some properties of this concept such as: the sum and product of two paranormal, the power of paranormal. Further, the relationships between the paranormal operators and other kinds of paranormal operators have been studied.

              In this paper, the packing problem for complete (  4)-arcs in  is partially solved. The minimum and the maximum sizes of complete (  4)-arcs in  are obtained. The idea that has been used to do this classification is based on using the algorithm introduced in Section 3 in this paper. Also, this paper establishes the connection between the projective geometry in terms of a complete ( , 4)-arc in  and the algebraic characteristics of a plane quartic curve over the field  represented by the number of its rational points and inflexion points. In addition, some sizes of complete (  6)-arcs in the projective plane of order thirteen are established, namely for  = 53, 54, 55, 56.

  The great scientific progress has led to widespread Information as information accumulates in large databases is important in trying to revise and compile this vast amount of data and, where its purpose to extract hidden information or classified data under their relations with each other in order to take advantage of them for technical purposes.

      And work with data mining (DM) is appropriate in this area because of the importance of research in the (K-Means) algorithm for clustering data in fact applied with effect can be observed in variables by changing the sample size (n) and the number of clusters (K)

The   study   of  torsion  {torsion   free)  fuzzy  modules   over   fuzzy

integtal  domain  as a generalization oftorsion (torsion free) modules.

The purpose of this paper is to introduce and study the concepts of fuzzy generalized open sets, fuzzy generalized closed sets, generalized continuous fuzzy proper functions and prove results about these concepts.

         In this paper, developed Jungck contractive mappings into fuzzy Jungck contractive and proved  fuzzy fixed point for some types of generalize fuzzy Jungck contractive mappings.

    In this paper further properties of the fuzzy complete a-fuzzy normed algebra have been introduced. Then we found the relation between the maximal ideals of fuzzy complete a-fuzzy normed algebra and the associated multiplicative linear function space. In this direction we proved that if  is character on Z then ker is a maximal ideal in Z. After this we introduce the notion structure of the a-fuzzy normed algebra then we prove that the structure, st(Z) is -fuzzy closed subset of fb(Z, ) when  (Z,  , , ) is a commutative fuzzy complete a-fuzzy normed algebra with identity e.

In this paper, we introduce and study the essential and closed fuzzy submodules of a fuzzy module X as a generalization of the notions of essential and closed submodules. We prove many basic properties of both concepts.