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I think it's not true that every module (over arbitrary ring) is the sum of indecomposable modules, but I can't find counterexample and literature about this problem. Can anyone help me?

Also I have similar question: is every abelian group is quasiisomorphic to the direct sum of strongly indecomposable abelian groups? The help in this question (counterexamples, literature) is also needed.

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  • $\begingroup$ I gave an example in math.stackexchange.com/questions/409548/… $\endgroup$ Jun 8, 2013 at 23:47
  • $\begingroup$ @JackSchmidt: It have to be an example, but I can't realize that any group decomposition in this case comes from ring decomposition. $\endgroup$
    – ptashek
    Jun 13, 2013 at 12:17

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There is such a counterexample in

L.Fuchs, Infinite Abelian Groups, Chap.XIII, Theorem 91.5:

"There is a countable group which has not any non-zero indecomposable direct summand."

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  • $\begingroup$ Thank you! But is it possible to give not such complicated example, maybe not countable? $\endgroup$
    – ptashek
    Jun 8, 2013 at 8:50
  • $\begingroup$ @ptashek: I don't know. If I find such an example, I will write to you. $\endgroup$ Jun 8, 2013 at 8:53

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