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My question is short and popular: how to define an infinite tensor product of modules over a ring?

So, there is an infinite set $I$ and $A$-modules $M_i$. I should understand what $\otimes_{i\in I} M_i$ is. Could you avoid categorical terms?

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Umm, how can you assert a question is popular? Shouldn't you let others be the judge of that? – Thomas Andrews Jun 22 '13 at 6:44
1 \\ Is easier if you take a commutative ring. – Frank Murphy Jun 22 '13 at 7:41

The most obvious definition is to define a multi-linear operator on the set $\prod_{i\in I} M_i$, and the define the tensor product as a universal module with a multi-linear map from $\prod_{i\in I} M_i$. You'd have to prove that the such a universal module existed.

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