Supplementing Radicals and Decompositions of Near-Rings

If is a radical of near-rings and is its supplementing radical, then (N)(N) N. We address the issue when (N) (N) = N holds. In the variety F of near-rings in which the constants form an ideal, the assignment c: N N_c is a hereditary Kurosh–Amitsur radical, _c is characterized in terms of distributors and criteria are given for the decomposition N = c(N) c(N). In the subvariety A of all abstract affine near-rings, assigning the maximal torsion ideal (N) is a hereditary Kurosh–Amitsur radical. If such near-rings N A satisfy dcc on principal right ideals, then N splits into a direct sum N = (N) (N) where the additive group of (N) is torsionfree and divisible. Dropping dcc on principal right ideals, an ``essential" decomposition result is proved.