Complementary radicals revisited (Q1333078)

The authors study relationships between complementary radicals in classes of not necessarily associative rings, in \(s\)-varieties and in abstract affine near-rings. Their main result (Theorem 3.9) deals with a class \(Q\) of simple idempotent rings in an \(s\)-variety. Let \(s(Q)\) be all subdirectly irreducible rings with heart in \(Q\), and let \(t(Q)\) be those with heart not in \(Q\). Then the upper radical \(Us(Q)\) is a supersolvable dual radical and it is the unique largest subclass \(U\) such that \(U \cap Q = 0\). Whereas the upper radical \(Ut(Q)\) is a subidempotent dual radical, and it is the largest subclass \(V\) such that the simple rings of \(V\) are in \(Q\). \(Us(Q)\) and \(Ut(Q)\) are mutually dual radicals.