Distributively generated rings and distributive modules

A module is said to be distributively generated if it is generated by distributive submodules. We prove that the endomorphism ring of a finitely generated projective right module over a right distributively generated ring is a right distributively generated ring. IfM is a module over a ringA andA/J(A) is a normal exchange ring, thenM is a distributive module⇔M is a Bezout module. Translated fromMatematicheskie Zametki, Vol. 68, No. 4, pp. 568–578, October, 2000.     

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     

Right radicals for right distributively generated near-rings

For right near-rings the left representation has always been considered the natural one. However, Hanna Neumann [6] constructed her right near-rings by writing the reduced free group on the left of the near-ring. In [2] and [8] Neumann's ideas are placed in a more general setting in the sense that right R-groups are used to define radical-like objects in the near-ring R. The right 0-radical ^r J ₀(R) and the right half radical ^r J _½(R) are introduced in [2] where it is shown that for distributively generated (d.g.) near-rings R with a multiplicative identity and satisfying the descending chain condition for left R-subgroups ^r J ₀(R) = J ₂(R), the 2-radical from left representation. In this article we introduce the right 2-radical, ^r J ₂(R) for d.g. near-rings and discuss some of its properties. In particular, we show that for all finite d.g. near-rings with identity J ₂(R) = ^r J ₂(R).     

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.