Maximal $R$-sequences in ideals

Question

If $\alpha_1,...,\alpha_s$ is a maximal $R$-sequence in an ideal $I$ ($R$ is commutative with unity), is this always true that $I⊆P$, where $P\in\operatorname{Ass} (\alpha_1,...,\alpha_s)$?

In case $R$ is Noetherian this is true because, by prime avoidance theorem, the inclusion $I⊆∪P$ implies the result. But, I do not know the answer in the non-Noetherian case. Thanks for any cooperation.

Answer 1

In this answer it is given an example of a non-noetherian ring $R$ with $\operatorname{Ass}(R)=\varnothing$. Then $I=(0)$ isn't contained in an associated prime for trivial reasons.