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I'm trying to think of an example of a ring $A$ and ideals $I$,$J$ s.t. $I \cup J$ is not an ideal.

And what is the smallest ideal containing $I$ & $J$?

Will $A \mathbb{Z}$, $I = 2\mathbb{Z}$, and $J = 4\mathbb{Z}$ work? and the smallest ideal containing them be $\mathbb{Z}$, the integers?

Can someone add some explanation as to why this works? Thanks.

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    $2\mathbb{Z} \cup 4\mathbb{Z}=2\mathbb{Z}$2012-05-29
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    $2\mathbf Z$ and $3\mathbf Z$ would work, though. [Check whether the union is closed under addition.]2012-05-29
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    Ironically, most choices of $I$ and $J$ would work with $A = \mathbb{Z}$; you had the misfortune of making one of the special choices that doesn't.2012-05-29

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