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In a commutative ring $R$ if $I,J,K$ are ideals, is $(I+J)\cap (I+K)=I+J\cap K$. I think this is true but cannot find it referenced anywhere. Working modulo $I$, both sides are just $J\cap K$ and since all ideals contain $I$, we can lift this relation to $R$? Am I thinking straight here? I am also a bit apprehensive since I cannot seem to make an element wise argument for the result.

I still cannot find any counterexamples. If anyone can suggest one it would be of great help. I am also unable to see the flaw in the above argument.

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    @MartinBrandenburg: I did check both $\mathbb{Z}$ and $k[x]$. Maybe I just chose the "wrong" examples.2012-03-13

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