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In general the union of two subrings is not a subring, and likewise for the union of two ideals. However, I was wondering if there exists an example of a union of ideals that is actually a subring but is not an ideal? I've tried looking around without any success so I imagine the answer is no.

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If $\{I_j\}_{j\in J}$ is a non-empty family of ideals such that $I:=\bigcup_{j\in J}I_j$ is closed under addition (as required for a subring), then $I$ is already an ideal.

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