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Let $\{R_i\}_{i\in I}$ a collection of $M$ modules. I know that if $\{e_1,...,e_m\}$ is a basis of $\oplus_{i=1}^nR_i$, then, $$x\in \oplus_{i=1}^nR_i\iff \exists ! a_1,...,a_m\in M: x=a_1e_1+...+a_me_m.$$

Now, I have a definition when $I$ is unspecified, i.e. $$x\in \oplus_{i\in I}R_i\iff x_i=0\text{ except for a number finite of $i$}.$$

Is there a correlation between those two definitions ?

MSE
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  • The first thing is the definition of being a basis and is unrelated to the direct sum (which may not even have a basis). – Tobias Kildetoft Sep 22 '16 at 08:25
  • To me a logical definition of $x\in \oplus {i\in I}R_i$ would be there is unique $x_i\in R_i$ s.t. $x=\sum{i\in I}x_i$. Would that make sense ? @TobiasKildetoft – MSE Sep 22 '16 at 08:35
  • You seem to be going about this in the opposite order of what is usual. We define the direct sum as a set, we don't usually write up what this means in terms of when something is an element of that set (because what sort of form would the element have to begin with anyway?). – Tobias Kildetoft Sep 22 '16 at 08:37
  • Think about the sum $\mathbb{Z}_6\cong\mathbb{Z}_3\oplus\mathbb{Z}_2$, does it make sense talk about a basis? Whatever it is it must be for both as they are isomorphic. – Zelos Malum Sep 22 '16 at 08:40
  • @ZelosMalum: What does mean $A\oplus B$ for groups ? – MSE Sep 27 '16 at 08:53

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