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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?

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The answer to your question is negative.

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 $I_i$.

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