Characterizalions in terms of
endomorphisms and quasi-endomorphisms are obtained for torsion free abelian
groups with the property that each pure subgroup of finite rank is a quasi-summand.
A group has this property if and only if its ring of endomorphisms with finite rank is
2-fold ct-transitive, and hence k-fold ct-transitive for every k. This property is
equivalent to complete decomposability for countable groups the type set of
which satisfies the maximum condition. A stronger version of transitivity is
required to describe separable groups the type set of which satisfies the
maximum condition; to insure generality, it is shown that the maximum
condition does not imply countability of the type set, a result of independent
interest.
