Near-Ring Radicals and Class Pairs

Abstract

For near-ring ideal mappings ρ1 and ρ2, we investigate radical theoretical properties of and the relationship among the class pairs (ρ1: ρ2), [Formula: see text] and (ℛρ2: ℛρ1). Conditions on ρ1 and ρ2 are given for a general class pair to form a radical class of various types. These types include the Plotkin and KA-radical varieties. A number of examples are shown to motivate the suitability of the theory of Hoehnke-radicals over KA-radicals when radical pairs of near-rings are studied. In particular, it is shown that [Formula: see text] forms a KA-radical class, where [Formula: see text] denotes the class of completely prime near-rings and [Formula: see text] the class of 3-prime near-rings. This gives another near-ring generalization of the 2-primal ring concept. The theory of radical pairs are also used to show that in general the class of 3-semiprime near-rings is not the semisimple class of the 3-prime radical.