Let $\{I_{\beta}\}_{\beta \in B}$ in $R$, with $\beta$ a infinite set. Is $$\sum_{\beta \in B} I_{\beta}=\{f_1+\cdots+f_r:f_j \in I_{\beta_j} \mbox{ for some } I_{\beta_j}\}\ ?$$ I feel confused because by this description, every element in $\sum_{\beta \in B} I_{\beta}$ is a finite sum, but $\sum_{\beta \in B} I_{\beta}$ is a infinite sum of ideals. Also I don't find a definition of $\sum_{\beta \in B} I_{\beta}$, can you tell me what the definition is?
2026-04-06 19:32:37.1775503957
The definition of the infinite sum of ideals.
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