I'm looking for any (obviously weaker) generalization of this famous theorem in the special case that the family of ideals is not finite.
2026-04-23 01:29:34.1776907774
Generalization of Chinese Remainder Theorem to infinite ideals
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Getting this out of unanswered-limbo
The key point of the CRT, surjectivity of the map, does not even work for $\mathbb Z$. This can be seen, as mentioned in the comments, by noting the product cannot is not Noetherian, but also because the product is uncountable.
I would say this does not have a reasonable generalization without further qualification.