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Suppose you have a semisimple ring $R$, and want to decompose it into a sum of simple left ideals. Let $\{L_i\}$ be a family of simple left ideals, such that no two are isomorphic, and any simple left ideal of $R$ is isomorphic to some $L_i$.

Then writing $R_i=\sum_{L\simeq L_i}L$, it follows that $R=\sum_{i\in I}R_i$, and so $1=\sum_{i\in I}e_i$ for $e_i\in R_i$. Apparently this sum is actually finite, but there is no explanation as to why. Why are almost all $e_i=0$ in this sum for $1$?

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    We don't seem to have a way of making sense of infinite sums. $\sum_{i \in I} R_i$ has to stand for the set of all $\sum_{i \in I} x_i$, $x_i \in R_i$ for all $i$, $x_i = 0$ for almost all $i$.2012-04-12
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    Oh, so the sum is finite just because it wouldn't make sense otherwise?2012-04-12

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