  Let $A$ be a Noetherian ring, $J\subseteq A$ an ideal and $C$ a finitely
generated $A$-module. In this note we would like to prove the following
statement. Let $\{I_n\}_{n\geq 0}$ be a collection of ideals satisfying : (i)
$I_n\supseteq J^n$, for all $n$, (ii) $J^s\cdot I_s \subseteq I_{r+s}$, for all
$r,s\geq 0$ and (iii) $I_n\subseteq I_m$, whenever $m\leq n$. Then
$\Ass_A(I_nC/J^nC)$ is independent of $n$, for $n$ sufficiently large. Note
that the set of prime ideals $\cup_{n\geq 1} \Ass_A(I_nC/J^nC)$ is finite, so
the issue at hand is the realization that the primes in $\Ass_A(I_nC/J^nC)$
\textit{do not} behave periodically, as one might have expected, say if
$\bigoplus _{n\geq 0}I_n$ were a Noetherian $A$-algebra generated in degrees
greater than one. We also give a multigraded version of our results.
