For an abelian group G and a
ring R,R is a ring on G if the additive group of R is isomorphic to G. G is nil if the
only ring R on G is the zero ring, R2= {0}.G is radical if there is a nonzero ring on
G that is radical in the Jacobson sense. Otherwise, G is antiradical. G is semisimple
if there is some (Jacobson) semisimple ring on G, and G is strongly semisimple if G is
nonnil and every nonzero ring on G is semisimple. It is shown that the only strongly
semisimple torsion groups are cyclic of prime order, and that no mixed group is
strongly semisimple. The torsion free rank one strongly semisimple groups
are characterized in terms of their type, and it is shown that the strongly
semisimple and antiradical rank one groups coincide. For torsion free groups it is
shown that the property of being strongly semisimple is invariant under
quasi-isomorphism and that a strongly semisimple group is strongly indecomposable.
Further, for a strongly indecomposable torsion free group G of finite rank, the
following are equivalent: (a) G is semisimple, (b) G is strongly semisimple,
(c) G≅R+ where R is a full subring of an algebraic number field K such
that [K,Q] = rank G where Q is the field of rational numbers and R = Jπ,
where π is either empty or an infinite set of primes in K,(d)G is nonnil and
antiradical.
