Generalized fuzzy ideals of near-rings

The concept of（∈,∈∨q）-fuzzy subnear-rings（ideals） of a near-ring is introduced and some of its related properties are investigated.In particular,the relationships among ordinary fuzzy subnear-rings（ideals）,（∈,∈∨ q）-fuzzy subnear-rings（ideals） and（∈,∈∨q）-fuzzy subnear-rings（ideals） of near-rings are described.Finally,some characterization of [μ]t is given by means of（∈,∈∨ q）-fuzzy ideals.     

Matrix Near-rings over Centralizer Near-rings

For a finite group G and a subgroup A of Aut(G), let M_A(G) denote the centralizer near-ring determined by A and G. The group G is an M_A(G)-module. Using the action of M_A(G) on G, one has the n × n generalized matrix near-ring Mat_n(M_A(G);G). The correspondence between the ideals of M_A(G) and those of Mat_n(M_A(G);G) is investigated. It is shown that if every ideal of M_A(G) is an annihilator ideal, then there is a bijection between the ideals of M_A(G) and those of Mat_n(M_A(G);G).1991 Mathematics Subject Classification: 16Y30     

On matrix near-rings

Let R be a right near-ring with identity and M_n(R) be the near-ring of n 2 n matrices over R in the sense of Meldrum and Van der Walt. In this paper, M_n(R) is said to be s\sigma-generated if every n 2 n matrix A over R can be expressed as a sum of elements of X_n(R), where X_n(R)={f_ij^r | 1\leqq i, j\leqq n, r ? R}X_n(R)=\{f_{ij}^r\,|\,1\leqq i, j\leqq n, r\in R\}, is the generating set of M_n(R). We say that R is s\sigma-generated if M_n(R) is s\sigma-generated for every natural number n. The class of s\sigma-generated near-rings contains distributively generated and abstract affine near-rings. It is shown that this class admits homomorphic images. For abelian near-rings R, we prove that the zerosymmetric part of R is a ring, so the class of zerosymmetric abelian s\sigma-generated near-rings coincides with the class of rings. Further, for every n, there is a bijection between the two-sided subgroups of R and those of M_n(R).     

On Prime and Semiprime Near-Rings with Generalized Derivation

Abstract

Let N be a left near-ring and S be a nonempty subset of N. A mapping F from N to N is called commuting on S if [F(x),x] = 0 for all x € S. The mapping F is called strong commutativity preserving (SCP) on S if [F(x),F(y)] = [x,y] for all x, y € S. In the present paper, firstly we generalize the well known result of Posner which is commuting derivations on prime rings to generalized derivations of semiprime near-rings. Secondly, we investigate SCP-generalized derivations of prime near-rings.     

The augmentation ideal in group near-rings

The definition of the group near-ring R[G] of the near-ring R over the group G as a near-ring of mappings from R ^(G) to itself is due to Le Riche et al. (Arch Math 52:132–139, 1989). In this paper we consider the augmentation ideal Δ of R[G]. If the exponent of G is not 2, then the structure of ΔR ^(G) is determined in terms of commutators and distributors. This is then used to show that Δ is nilpotent if and only if R is weakly distributive, has characteristic p ⁿ for some prime p and G is a finite p-group for the same prime p.      

Centers and Generalized Centers of Near-Rings

Let N be a right near-ring. Denote by C(N) the multiplicative center of N, and by N _d the set of left-distributive elements of N. In general, C(N) need not be closed under the addition of N. However, the generalized center of N, GC(N) = {a ? N|an _d  = n _d a for all n _d  ? N _d }, is always a subnear-ring of N containing C(N). In this article, we study the problem of determining when C(N) is a subnear-ring of N. We show that, for certain classes of near-rings, C(N) is a subnear-ring of N if and only if C(N) = GC(N). Examples are given to show the limits of the theory.     

Medial near-rings

In this paper we discuss (left) near-rings satisfying the identities:abcd=acbd,abc=bac, orabc=acb, called medial, left permutable, right permutable near-rings, respectively. The structure of these near-rings is investigated in terms of the additive and Lie commutators and the set of nilpotent elementsN (R). For right permutable and d.g. medial near-rings we obtain a Binomial Theorem, show thatN (R) is an ideal, and characterize the simple and subdirectly irreducible near-rings. Natural examples from analysis and geometry are produced via a general construction method.     

On a Question of J. H. Meyer

Abstract

In this paper, we present a partial solution to the following question of J. H. Meyer (Meyer, J. H. (1986). Matrix Near-Rings. Ph.D. dissertation, University of Stellenbosch, South Africa, Prob. 11): Find a necessary and sufficient condition for a near-ring to be isomorphic to a matrix near-ring. In fact, three characterization results for abstract affine matrix near-rings are given. As a corollary, we get that, for each n ≥ 2, the class of all n × n matrix near-rings over abstract affine near-rings is finitely axiomatizable.     

一般欧氏空间上的广义规范算子与广义规范矩阵

研究了一般欧氏空间上的广义规范算子与广义规范矩阵的性质。