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If a reduced torsion abelian group has no cyclic direct summands of order greater than 2, is it an elementary abelian 2-group?

Background: I'm trying to classify the groups whose group rings have a certain property related to a class of rings that I'm studying. It can be shown that no group with a cyclic direct summand of order greater than 2 has this property. I'd like to conclude that the only reduced torsion abelian groups with this property are the elementary abelian 2-groups. I know from Kaplansky's book that every reduced torsion abelian group has a finite cyclic direct summand (Theorem 9). I'm having trouble getting from there to the conclusion I'd like.

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Yes; the key is Lemma 65.4 in Fuchs's "Infinite Abelian Groups", volume II (which basically shows every element has order 2). – user641 Jun 23 '11 at 15:50

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