Matrix near-rings and 0-primitivity

In this article, we study the implication of the primitivity of a matrix near-ring ${\mathbb{M}_n(R) (n >1 )}${\mathbb{M}_n(R) (n >1 )} and that of the underlying base near-ring R. We show that when R is a zero-symmetric near-ring with identity and \mathbbM_n(R){\mathbb{M}_n(R)} has the descending chain condition on \mathbbM_n(R){\mathbb{M}_n(R)}-subgroups, then the 0-primitivity of \mathbbM_n(R){\mathbb{M}_n(R)} implies the 0-primitivity of R. It is not known if this is true when the descending chain condition on \mathbbM_n(R){\mathbb{M}_n(R)} is removed. On the other hand, an example is given to show that this is not true in the case of generalized matrix near-rings.     

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.     

D-strong and almostD-strong near-rings

In this paper,D-strong and almostD-strong near-rings have been defined. It has been proved that ifR is aD-strongS-near ring, then prime ideals, strictly prime ideals and completely prime ideals coincide. Also ifR is aD-strong near-ring with identity, then every maximal right ideal becomes a maximal ideal and moreover every 2-primitive near-ring becomes a near-field. Several properties, chain conditions and structure theorems have also been discussed.Most of the parts of this paper are included in author's doctoral dissertation at Sukhadia University Udaipur (1983). The author expresses his gratitude to Dr.S. C. Choudhary for his kind guidance.     

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.     

Ultraproducts and ultra-limits of near-rings

The construction of ultraproducts, ultrapowers and ultralimits have proven useful, both for logic and for special (algebraic) structures. It seems that these constructions might be especially useful in studying near-rings; they might provide solutions to long-standing problems. Among other results, we show that ultraproducts of primitive near rings are again primitive.Part of this work was done while the second author was a Visiting Professor at the University of Southwestern Louisiana (Lafayette, La). This author expresses his gratitude for the received hospitality and for most valuable discussions at this Department. Special thanks are due to Professors D. Blumberg, H. Heatherly and A. Iskander.     

Medial and distributively generated near-rings

In this paper we note some properties of fully invariant additive subgroups of near-rings and apply these results to d.g., medial, or subdirectly irreducible near-rings     

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.