Proposition

Give an example of a ring $A$ and ideals $I, J$ such that $I \cup J$ is not an ideal; in your example, what is the smallest ideal containing $I$ and $J$?

Solution

Let $A = \mathbb{C}[x], I = (x + 1), J = (x)$.

$I \cup J$ contains $x$ and $x + 1$. Since an ideal must be closed under subtraction, $x + 1 - x = 1$ must be in $I \cup J$. However, $1 \notin I$ and $1 \notin J$.

The smallest ideal containing $I$ and $J$ is $A$ because any ideal containing $I$ and $J$ must contain $1$, and the only ideal containing $1$ is $A$.