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I'm trying to understand this proof

The only thing I didn't understand is why there exists a finite subset $L$ such that $1_R=\sum_{l\in L}i_l$.

It should be a silly doubt, I'm sure I'm forgetting something.

Thanks.

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The point is that for $\{I_\lambda\}_{\lambda\in\Lambda}$ a set of ideals, the ideal

$\sum_{\lambda\in\Lambda} I_\lambda$

is the set of all finite sums $i_1+\ldots+i_n$ where each element lies in some $I_\lambda$. We don't include infinite sums because you can't make sense of an infinite sum in arbitrary ring. Thus, since $1\in \sum_{\lambda\in\Lambda} I_\lambda$, it must be expressible as a finite sum of elements, and so we can take just finitely many of the $I_\lambda$.

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