Let $R$ be a commutative ring with identity and let $I$ be an ideal of $R$. I want to show that there is an isomorphism of groups $R/I^2 \cong R/I \oplus I/I^2$. I think that this should follow from some isomorphism theorem or Chinese remainder theorem, but I haven't been able to get it.
2026-03-31 19:44:49.1774986289
Isomorphism of groups involving quotients
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The truth is that this claim is not true.
Example: $R = \mathbb Z$, $I = 2\mathbb Z$.
On the left you have $\mathbb Z / 4 \mathbb Z$, which is a cyclic group.
On the right you have $\mathbb Z / 2 \mathbb Z \oplus \mathbb Z / 2 \mathbb Z$, which is not cyclic.