PRIMITIVITY IN NEAR-RINGS WITH LOCALIZED DISTRIBUTIVITY CONDITIONS

Abstract

Let K, S, T be subsets of a near-ring R. Then K is (S, T)-distributive if: s(k ₁ + k ₂)t = sk ₁ t + sk ₂ t, for each k ₁, k ₂ ε K, s ε S, t ε T; and K is (S, T)-d.g. on X if K is (S, X)-distributive and T is contained in the additive subgroup generated by X. This paper considers υ-primitivity and the associated ℑ_υ radicals under various such conditions, particularly where S, T, and K are powers of R. Natural examples which illustrate and delimit the theory are given.