Let $R$ be a ring with unity. Is possible to have $M\oplus M \simeq N \oplus N$ and $M$ not isomorphic to $N$, where $M$ and $N$ are $R$-modules?
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3This is false, it was answered here http://mathoverflow.net/questions/24697/isomorphism-between-direct-sum-of-modules – user353673 Aug 30 '16 at 23:02
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1@user353673 Consider posting an answer! – Caleb Stanford Aug 30 '16 at 23:03
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A good question at this point would be: does there exist a counterexample for every ring $R$ (unital or not)? Also, we can restrict to the discussion of unitary modules in case $R$ is unital (but we can also look at non-unitary modules as well). – Batominovski Aug 30 '16 at 23:15
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1See this example, and try to use Search before posting. – user26857 Aug 31 '16 at 08:30
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See also http://math.stackexchange.com/questions/27744/does-g-oplus-g-cong-h-oplus-h-imply-g-cong-h-in-general?noredirect=1&lq=1 – Watson Aug 31 '16 at 08:53
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The following MathOverflow thread discusses the question and several counterexamples are presented. Perhaps there are even more elementary ones?
If $M$ and $N$ are finitely generated and $R$ is a principal domain, then the assertion is true. This was discussed in the following thread.
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