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 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).     

Generalized matrix near-rings

Let R be a right near-ring with identity. The k×k matrix near-ring over R, Mat_k(R R), as defined by Meldrum and van der Walt, regards R as a left mod-ule over R. Let M be any faithful left R-module. Using the action of R on M, a generalized k×k matrix near-ring, Mat_k(R M), is defined. It is seen that Mat_k(R M) has many of the features of Mat_k(R R). Differences be-tween the two classes of near-rings are shown. In spe- cial cases there are relationships between Mat_k(R M) and Mat_k(R R). Generalized matrix near-rings Mat_k(R M) arise as the “right near-ring” of finite centraiizer near-rings of the form M _A{G)> where G is a finite group and A is a fixed point free automorphism group on G.     

On equiprime near-rings

An equiprime near-ring is a generalization of prime ring. Firstly some axioma-tics concerning equiprime near-rings are discussed, e.g. their relation to the other notions of primeness for near-rings, primitive near-rings and near-fields. Secondly we investigate the equiprimeness of some well-known examples of near-rings.     

Radicals of regular near-rings

Analogues of ring theory results concerning the Jacobson radical of a regular ring are obtained for near-rings with a two-sided zero. The quasiradical and the radical-subgroup of a regular near-ring are shown to be {0}. Some sufficient conditions are obtained for the radical and the primitiveradical of a regular near-ring to be {0}. Necessary and sufficient conditions are determined for a near-ringR which satisfies d. c. c. onR-subgroups ofR to be regular.