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Singly cogenerated annihilator classes

1987; Elsevier BV; Volume: 109; Issue: 1 Linguagem: Inglês

10.1016/0021-8693(87)90168-2

1090-266X

Manfred Dugas, Temple H. Fay, Saharon Shelah,

Finite Group Theory Research

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 pλ-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.