Vol. 53, No. 1, 1974

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Completely decomposable groups which admit only nilpotent multiplications

Charles Irvin Vinsonhaler and William Jennings Wickless

Vol. 53 (1974), No. 1, 273–280

A triangle of size n is a collection {Au} 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(Au) + T(As) T(Aus) for all consecutive sequences u,s. Here T(Av) denotes the type of the rank one torsion-free abelian group Av. If A = iIAi is a direct sum of rank one torsion-free abelian groups Ai, let Δ(A) = sup{n|∃ a triangle of size n of groups chosen, possibly with repetitions, from {Ai|i I}}, Δ(A) = sup{n|∃ a triangle of size n of groups chosen without repetition from {Ai|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).

Mathematical Subject Classification 2000
Primary: 20K20
Received: 5 February 1973
Revised: 30 January 1974
Published: 1 July 1974
Charles Irvin Vinsonhaler
William Jennings Wickless