On Division Near-Rings

Abstract

The following results (9, Exercise 26, p. 10; 1, Theorem 9.2; 8, Theorem III. 1.11) are known.(A) Let R be a ring with more than one element. Then R is a division ring ifand only if for every a ≠0 in R, there exists a unique b in R such that aba = a.(B) Let R be a near-ring which contains a right identity e ≠ 0. Then R is adivision near-ring if and only if it contains no proper R-subgroups.(C) Let R be a finite near-ring with identity. Then R is a division near-ringif and only if the R-module R+ is simple.In this paper we will show that (A) can be generalized to distributively generated near-rings. We also will extend (B) and (C) to a larger class of near-rings.