# When does the prime avoidance lemma hold with sum instead of union?

The following result is known as prime avoidance lemma:

Given $I_i$ a list of $n$ prime (except perhaps two of them) ideals of a conmutative ring $R$ and suppose that $L \subseteq \cup I_i$ then $\exists k. L \subseteq I_k$.

This result does not behave nicely with the lattice of ideals since the supremum is the sum of ideals, not the union. Under what conditions can I substitute the union in the above result by the sum of ideals?

The answer is negative in almost any cases. It works if and only if the ideals on the RHS are all contained in one of them, i.e. the sum coincides with the union. In all other cases, you can take $L$ equal to the sum for a counterexample.