Let R = $$\prod_{i\in \Delta }Z_{i} $$ where $$Z_{i} $$ Rings of integer It's obvious that the product of a rings is itself a ring And in finite case the ideal of the product = Product of Ideals that is if I is an ideal of R , then $$I = \prod_{i\in \Delta }I_{i}$$ and $$\Delta $$ is finite now the question is what if $$ \Delta $$ is infinite can we still have $$I = \prod_{i\in \Delta }I_{i}$$ ?
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1No, in general there are many more ideals. – Qiaochu Yuan Oct 11 '20 at 18:13
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See, for example: https://math.stackexchange.com/a/1563190/232 – Qiaochu Yuan Oct 11 '20 at 18:19