Let 𝒜 denote the class of
torsion free Abelian groups of finite rank. It is shown that for A ∈𝒜, there is a
quotient divisible subgroup QD(A) such that A∕QD(A) is a reduced torsion group.
Furthermore, QD(A) and A∕QD(A) are unique up to quasi-isomorphism. Let ℬ
denote the subclass of 𝒜 of groups A such that for almost all primes p, the p-primary
component of A∕QD(A) is the direct sum of rp(A) isomorphic cyclic groups
where rp(A) denotes the p-rank of A. The groups in ℬ are classified up to
quasi-isomorphism, which generalizes the Beaumont-Pierce classification of quotient
divisible groups.
The main results of this paper concern the subclass g of 𝒜 of groups A such that
rp(A) ≦ 1 for all primes p. The class ℰ may be profitably treated as a generalization
of the class of rank one groups in 𝒜.
