Completely decomposable groups which admit only nilpotent multiplications

Abstract

A triangle of size n is a collection {A u } of n(n + l)/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 lii + jn, satisfying T{A U ) + T(A S ) ^ T(A US ) for all consecutive sequences u, s. Here T(A ) denotes the type of the rank one torsion-free abelian group A v . If A i e r A is a direct sum of rank one torsion-free abelian groups A if let (A)sup {n | 3 a triangle of size n of groups chosen, possibly with repetitions, from {A t \ is I}}, '{A) -sup{%| 3 a triangle of size n of groups chosen without repetition from {Ai I iel}}. An abelian group (G, +) is radical iff whenever (R, +, ) is a ring with (R, +) ^ (G, +) there exists a positive integer n with R n -(0).