Abstract
If ϱ is a radical of near-rings and ϱ is its supplementing radical, then ϱ(N)⊕ϱ(N) ⊲ N. We address the issue when ϱ(N) ⊕ϱ(N) = N holds. In the variety F of near-rings in which the constants form an ideal, the assignment c: N → Nc 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 τ(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 = τ(N) ⊕τ(N) where the additive group of τ(N) is torsionfree and divisible. Dropping dcc on principal right ideals, an ``essential" decomposition result is proved.
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