The Artin-Rees lemma states that if $R$ is a Noetherian ring, $I \subseteq R$ is an ideal and $N \subseteq M$ are finitely generated $R$-modules, then there exists an integer $k$ such that for every $n \ge k$, we have $$I^n M \cap N = I^{n-k} \cdot (I^kM \cap N) \subseteq I^{n-k} N \, .$$

The inclusion $I^n M \cap N \subseteq I^{n-k}N$ is sometimes called the weak Artin-Rees-Lemma.

There exist several versions of this weak Artin-Rees Lemma which are uniform with respect to $I$, i.e., the corresponding integer $k$ depends only on the modules $N \subseteq M$ of $R$ and works for every ideal $I$ of $R$.

Do there exist variants of this result which are also uniform with respect to the modules $N$ and $M$?

I wonder whether it is possible to bound the corresponding $k$'s in terms of invariants of the ring $R$. For example, suppose that $R$ is a regular local ring. Is $k$ bounded by the Krull dimension of $R$?