Published online by Cambridge University Press: 09 April 2009
A class (called c-radicals) is defined such that, given a c-radical P, there is in any class M′ a certain internal criterion that its upper radical UM′ = P, For P a non-c-radical (called a q-radical) there exists no smallest class M such that UM = P, and P is a q-radical if and only if for some M with P = UM there exists 0 ≠ R ∈ M such that when an image ¯ of R has a non-zero image in M there exists an infinite chain of epimorphisms ¯ → R1 → R2 → … with all R1 ∈ M and no Ri, the image of any Rj, with j > i. Several examples of such rings are constructed including a ring all of whose images are primitive. Thus all radicals contained in the Jacobson radical are q-radicals.