The cancellation of torsion abelian groups in direct sums

Abstract

J6nsson and Tarski [5] proved that any finite abelian group has the cancellation property,l and, in answer to a question of Kaplansky [6], Cohn [I] and Walker [8] extended this result to finitely generated abelian groups. More recently, Rotman and Yen [7] showed that a countably generated torsion module (over a complete discrete valuation ring) all of whose Ulm invariants are finite, can be cancelled from isomorphic direct sums provided that the complementary summands are countable generated modules of finite rank. Here a description is given of those countable torsion abelian groups having the cancellation property. Thus: