We consider the determination of ring radicals by classes of modules as first discussed by Andrunakievich and Ryabukhin, but
in cases where the modules concerned are defined by additive properties. Such a radical is the upper radical defined by the
class of subrings of a class of endomorphism rings of abelian groups and is therefore strict. Not every strict radical is
of this type, and while the A-radicals are of this type, there are others, including some special radicals. These investigations bring radical theory into
contact with (at least) two questions from other parts of algebra. Which rings are endomorphism rings? For a given ring R, which abelian groups are non-trivial R-modules?
It is shown that in a well known characterization of radical classes, closure under unions of chains of ideals can be replaced
by closure under unions of continuous well-ordered chains of ideals. Some consequences are discussed.