Let ⟨G+⟩ be an abelian group.
A ring R with additive group ⟨R+⟩ isomorphic to ⟨G+⟩ is a ring on G. G is nil
(radical) if and only if R^{2} = (0)( R is nilpotent) for all rings R on G. It is shown that
G is a mixed radical group if and only if T is divisible and G∕T is radical, where T
is the maximal torsion subgroup of G. Thus, the study of radical groups
is reduced to the torsion free case. A torsion free group G is of field type
if and only if there exists a ring R on G such that Q ⊗ R is a field. It is
shown that a torsion free group of finite rank is radical if and only if it has
no strongly indecomposable component of field type. It follows that finite
direct sums of finite rank radical groups are radical. If G is torsion free an
element x ∈ G is of nil type if and only if the height vector h(x) = ⟨m_{x}⟩ is
such that 0 < m_{i} < ∞ for infinitely many i. Multiplications on torsion free
groups all of whose nonzero elements are of nil type are discussed under the
assumption of three chain conditions on the partially ordered set of types.
Two special classes of rank two torsion free radical groups are characterized.
An example is given of a torsion free radical group homogeneous of nonnil
type, and a simple condition is given for such a homogeneous group to be
nonradical.
