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Does there exist a nonzero $R$-module $M$ so that $M \cong M\oplus M$, where here $R$ is any ring with unity?

I'm not sure if this is true; I don't think it's true but I can't prove that. Clearly $M$ must be infinite for this to be true.


marked as duplicate by user26857 abstract-algebra Oct 23 '16 at 7:29

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  • $\begingroup$ The trivial module $\endgroup$ – Zelos Malum Oct 23 '16 at 1:41

One example is a vector space of infinite dimension $\kappa$, since the direct sum will have a basis of cardinality $\kappa + \kappa = \kappa$.

  • 1
    $\begingroup$ +1 (we can also replace "vector space" with "free $R$-module") $\endgroup$ – user144221 Oct 23 '16 at 1:43
  • $\begingroup$ Is there any easier example you know of? This was intended to be a basic book problem after an introductory section on modules. $\endgroup$ – mathworker21 Oct 23 '16 at 4:16

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