Union of three incomparable nonprime ideals
= Union of three incomparable nonprime ideals
In $\mathbb F_2[u,v]/(u,v)^2$, the <ideals> $(u)$, $(v)$ and $(u+v)$ are pairwise incomparable and nonprime. Their union is the <ideal> $(u,v)$: the three lines cover its two-dimensional <vector space> over $\mathbb F_2$. This shows why a primality hypothesis matters in <prime avoidance>.