Example of ideals abstract algebra

Give an example of a ring A and ideals I,J such that I U J is not an ideal.

In your example, what is the smallest ideal containing I and J?

Let A = Z, the ring of integers, and consider I = <2>, and J = <3>.

I U J is not closed under addition, because 2 + 3 = 5 is not an element of either ideal.
Hence, I U J is not an ideal of Z.

For this example, <6> is the smallest ideal containing <2> and <3>.

I hope this helps!