Let 𝒜 be the category of
Abelian groups, let ℬ be the class of bounded Abelian groups, and form the quotient
category 𝒜∕ℬ. The principal goal of this paper is a complete set of invariants for
direct sums of countable reduced p-groups, such groups considered as objects of
the category 𝒜∕ℬ. Specifically, it will be shown that two direct sums of
countable reduced p-groups G and H are isomorphic in 𝒜∕ℬ if and only if
there is an integer k ≧ 0 such that for all ordinal numbers α and all integers
r ≧ 0
and
where fG(β) and f1J(β) denote the β-th Ulm invariants of G and H, respectively.
Thus a complete set of 𝒜∕ℬ-isomorphism invariants for such groups is an equivalence
class of Ulm invariants, the equivalence relation being given by these two
inequalities.
