Completely decomposable groups which admit only nilpotent multiplications

A triangle of size n is a collection {A_u} of n(n + 1)∕2 (not necessarily distinct) rank one torsion-free abelian groups indexed by all integer sequences of the form u = i,i + 1,⋯ , i + j with 1 ≦ i ≦ i + j ≦ n, satisfying T(A_u) + T(A_s) ≦ T(A_us) for all consecutive sequences u,s. Here T(A_v) denotes the type of the rank one torsion-free abelian group A_v. If A = ⊕_i∈IA_i is a direct sum of rank one torsion-free abelian groups A_i, let Δ(A) = sup {n|∃ a triangle of size n of groups chosen, possibly with repetitions, from {A_i|i ∈ I}, Δ′(A) = sup {n|∃ a triangle of size n of groups chosen without repetition from {A_i|i ∈ I}. An abelian group (G,+) is radical iff whenever (R,+,⋅) is a ring with (R,+)≅(G,+) there exists a positive integer n with Rⁿ = (0).