# Can we have a Primary Avoidance Theorem ?

Prime Avoidance Theorem says:

Let $P_1, P_2,\dots, P_n$ be prime ideals in a commutative ring $R$ and let $I$ be an ideal of $R$ such that $I \subseteq P_1 \cup P_2 \cup \cdots \cup P_n$. Then $I \subseteq P_k$ for some $k\in \{1,2,\dots,n\}$.

Is true if replace prime ideals by primary ideals?

Consider a local artinian ring $$R$$ and $$I_1,I_2,I_3$$ three ideals with $$I_k\not\subseteq I_l$$ for $$k\neq l$$ and such that $$I=I_1\cup I_2\cup I_3$$ is also an ideal of $$R$$. Then it's clear that $$I$$ can't be contained in one of the $$I_i$$s.
For a concrete example take $$R=\mathbb F_2[X,Y]/(X,Y)^2$$ and $$I=(x,y)$$. I leave you the pleasure to find out the three ideals $$I_i$$.