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.