Singly cogenerated annihilator classes

We concern ourselves with the question of when an annihilator class (a class closed under formation of arbitrary products and subgroups) of abelian groups is cogenerated by a single group. We show that the class of p^λ-reduced groups (where p is a prime and λ is an ordinal greater than ω) is not singly cogenerated. If G is a cotorsion-free group, we show that the torsion-free class cogenerated by G is not singly cogenerated as an annihilator class. This result permits identification of all singly cogenerated annihilator classes which are also closed under formation of extensions, and so we characterize those singly generated radicals which are idempotent. They are precisely the radicals determined by annihilator classes singly cogenerated by a pure injective.