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 near-rings with ATM

A near-ring (N, +, ·) has an almost trivial multiplication (ATM) if the product of two elements belongs to the intersection of the additive cyclic groups generated by these two elements. We give examples of such near-rings and we show that the prime radical has a strong influence on the structure of the additive group of a near-ring with ATM.     

On annihilators of polynomial near-rings

Let R be a ring and let R ₀[x] be the polynomial near-ring over R. We study relations between the set of annihilators in R and the set of annihilators in R ₀[x].     

On arithmetical varieties of near-rings

This research was done during the second aurthor's stay in National Cheng-Kung University, Tainan. Support by the grant No. VRP92034 from the National Science Council of ROC is gratefully acknowledged.     

On zero-divisors of near-rings of polynomials

In this paper, we are interested to study zero-divisor properties of a 0-symmetric nearring of polynomials R₀[x], when R is a commutative ring. We show that for a reduced ring R, the set of all zero-divisors of R₀[x], namely Z(R₀[x]), is an ideal of R₀[x] if and only if Z(R) is an ideal of R and R has Property (A). For a non-reduced ring R, it is shown that Z(R₀[x]) is an ideal of Z(R₀[x]) if and only if ann_R({a, b}) ∩ N i?(R) ≠ 0, for each a, b ∈ Z(R). We also investigate the interplay between the algebraic properties of a 0-symmetric nearring of polynomials R₀[x] and the graph-theoretic properties of its zero-divisor graph. The undirected zero-divisor graph of R₀[x] is the graph Γ(R₀[x]) such that the vertices of Γ(R₀[x]) are all the non-zero zero-divisors of R₀[x] and two distinct vertices f and g are connected by an edge if and only if f ? g = 0 or g ? f = 0. Among other results, we give a complete characterization of the possible diameters of Γ(R₀[x]) in terms of the ideals of R. These results are somewhat surprising since, in contrast to the polynomial ring case, the near-ring of polynomials has substitution for its “multiplication” operation.     

On Jacobson type radicals of near-rings

The main result of this paper is a negative answer to the problem concerning the radicalness in the sense of Kurosh—Amitsur of the near-ring radicalJ ₀. We shall give negative answers to some hereditariness problems of near-ring radicals too.     

On the summands of near-rings with ATM

We say that a near-ring (N,+,·) has an almost trivial multiplication (ATM) if the product of two elements belongs to the intersection of the additive cyclic groups generated by these two elements. We show that every finite near-ring with ATM can be decomposed to a direct sum where the summands are either near-rings defined on cyclic groups or near-rings whose minimal ideals are zero near-rings. Finally, we show how to construct these summands on cyclic groups.     

On commutativity of near-rings with generalized derivations

In this paper we prove some theorems of commutativity for near-rings with generalized derivations. As a consequence of the results obtained, we generalize some published results. Also, we give some examples to show that some conditions in some results obtained are not redundant.     

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.