A theorem on compatible N-groups

Abstract

A near-ring N is a set N with binary operations + and · satisfying the conditions (1) (N, +) is a group, (2) (N, ·) is a semigroup, and (3) · satisfies one of the distributive laws over +. (N, +) need not be an abelian group and if the left distributive law holds, i.e. a · (b + c) = a · b + a · c for all a, b, c ∈ N, then N is called a left near-ring. Similarly, the notion of a right near-ring may be defined.