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Is there a difference between saying that the direct sum of $|S|$ copies of a ring $R$ is the set of all functions $f: S \to R$ such that they are zero except in finitely many places and saying that $\oplus_{s \in S} R$ is the set of all tuples $(r_s, r_{s^\prime}, r_{s^{\prime \prime}}, \dots )$ such that only finitely many $r_s$ are non-zero?

It seems to be the same and the second seems more intuitive but the first one is used on Wikipedia for some reason.

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    @ClarkKent: Yes, $(r_s)_{s\in S}$, which gives the tuple as a family, would be a reasonable notation. Of course, then you are coming closer to the identification of a "tuple"/family with a function, since a family is just a function with domain the index set.2012-04-22

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The "tuples" characterization is the same thing unless you think there's no such thing as a tuple $(\ldots,\bullet, \bullet,\bullet,\ldots)$ when the number of components is uncountable. I.e. the "tuples" way of saying it seems to assume the number of copies you're summing is at most countably infinite.

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    Well, this is all really a matter of how you co$n$strue the $n$otation. But the "functions" characterization doesn't seem to have the same potential for confusion about the meaning. Maybe the "tuples" characterization is more intuitive if you're accustomed to it.2012-04-22
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There is no essential difference. You may consider the way of expressing a direct product $\prod_{\alpha\in I} A.$ We may define it as the set of functions from $I\rightarrow A$ because to every $\alpha\in I$ it uniquely associates an element in $A$. The direct sum construction you quoted is a special case of this.

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    It's not really a special case, because the direct sum is a proper subobject of the direct product in this context!2012-04-21