For near-ring ideal mappings p1 and p2, we investigate radical theoretical properties of and the relationship among the class pairs (p1: p2), (Sp2: Sp1) and (Rp2:Rp1). Conditions on p1 and p2 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 (pc: P3) forms a KA-radical class, where Pc denotes the class of completely prime nearrings and P3 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.