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I was reading about the construction of a tensor product of any right $R$-module $M$ and left $R$-module $N$ over a ring $R$.

Why is it required that $M$ and $N$ be right and left modules, respectively? How does the construction not work otherwise?

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    Well, if they were both left modules, say, you'd get stuff like $r(sx)\otimes y = sx \otimes ry = x \otimes s(ry) = x \otimes (sr)y = (sr)x \otimes y$, which is problematic if $R$ is not commutative.2012-03-08

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