Suppose that we have a Noetherian domain $R$ and two ideals $I$ and $J$ of $R.$
Now consider the minimal (or irredundant) primary decompositions $I=\bigcap\limits_{i=1}^r Q_i$ and $J=\bigcap\limits_{i=1}^s Q'_i$ of $I$ and $J$ respectively. If $I\supseteq J,$ are there any relations between the ideals appearing in the decompositions of $I$ and $J$ and the numbers $s$ and $r$?
For example if $R$ is a Dedekind domain, when we write an ideal as a product of prime ideals, we are able to obtain relations between the exponents of the prime ideals appearing in the decomposition of $I$ and $J.$
Thanks in advance.
As Youngsu mentioned above, there is in general no relation between the primary components of $I$ and $J$, or the numbers $r$ and $s$ (and Youngsu has given the example of $I = m$ in a local ring for a case where $r < s$).
For an example with $r > s$: take $R = k[x,y]$, $I = (x) \cap (x,y)^2 = (x^2,xy) \supseteq J = (x^2)$.
On a positive note, there are relations among the radicals of the primary components, i.e. the associated primes of $I$ and $J$. If $I = \bigcap_{i=1}^r Q_i \supseteq J = \bigcap_{i=1}^s Q_i'$, then writing $P_i = \sqrt{Q_i}$, $P_i' = \sqrt{Q_i'}$ yields
$$\bigcap_{i=1}^r P_i = \sqrt{I} \supseteq \sqrt{J} = \bigcap_{i=1}^s P_i'$$
Thus for every $i = 1, \ldots, r$, there exists $j$ such that $P_i \supseteq P_j'$ (in fact we can always choose $P_j' \in \text{Min}(J)$). More information can also be obtained from the short exact sequence $0 \to I/J \to R/J \to R/I \to 0$, which implies e.g. that every associated prime of $J$, that is not associated to $I/J$, is in fact an associated prime of $I$.