Looking for an example of an ideal contained in the union of other ideals, but not in any ideal individually

Question

I'm looking for an example of the following scenario:

$A, B, C $ are three ideals such that $C\subseteq A\cup B $ but $C\not\subseteq A $ and $C\not\subseteq B$.

Any help would be great!

Answer 1

This situation is impossible. Indeed, suppose you have such ideals $A,B$ and $C$. Since $C\nsubseteq A$, there is an element $b\in C$ that is not in $A$; thus it must be in $B$. Similarly, there is an element $a\in C$ that is not in $B$, so it must be in $A$.

Consider the element $a+b\in C$. This element is not in $A$, otherwise so would $b$. Neither is it in $B$, else so would $a$. Thus $a+b\notin A\cup B$, contradicting the fact that $C\subseteq A\cup B$.

Note that we only used the group structure of $A,B$ and $C$, not their ideal structure.