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 $1$?