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This question already has an answer here:

I've just assumed that this is false, since the problem statements says to compare it to a previous problem where $\{ a+b \ | \ a\in I, b\in J \}$ is ideal.

However, by trial and error I can't find two ideals where this doesn't hold.

Is this false, and if so what's the counterexample? I haven't found one thus far.

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marked as duplicate by rschwieb abstract-algebra Mar 18 '16 at 10:35

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

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Consider (x,y) and (z,w) in Z[x,y,z,w]. Then your set isn't closed under addition: it contains xz and yw but not xz+yw.

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    $\begingroup$ this works if we take $(x,y)$ twice and consider $x^2+y^2$. $\endgroup$ – Jorge Fernández Hidalgo Mar 17 '16 at 23:20
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    $\begingroup$ Or $I=J=\langle 2,X\rangle$ and consider $4+X^2$. $\endgroup$ – Henning Makholm Mar 17 '16 at 23:24

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