I have a simple question: Let $R=\mathbb{C} \lbrace t,u \rbrace$ be the ring of formal series in two variables. Let $I,J \subset R$ ideals of heigth one, and $I \subset J$. What is the relation of the primary decompositions of $I$ and $J$?

More precisely, when is true that $Ass(R/I) = Ass(R/J)$? Or at least one is contained in the another one?

For example, let be $I=(t^{3}+ut^{2}, t^{4})$ and $J=(t^{2})$, we have that $I \subset J$, and in this case $Ass(R/J)=\lbrace (t) \rbrace \subset Ass(R/I)=\lbrace (t), (t,u) \rbrace$.