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Usually for modules $M_1,M_2,N$ $$M_1 \times N \cong M_2 \times N \Rightarrow M_1 \cong M_2$$ is wrong. I'm just curious, but are there any cases or additional conditions where it gets true?

James B.

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    Hm... if $M_1$, $M_2$ and $N$ are free and finitely generated over a ring with the IBN, it holds true.2012-06-14
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    This is a **very hard** question: For example, let $R=k\[x_1,\ldots,x_n\]$. If $P$ is projective module of finite rank and $P\oplus R^n\cong R^{n+m}$, then $P\cong R^m\ \ldots$ But this is the Serre's Conjecture!2012-06-14

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