Let $R$ be a commutative ring, $I$ is an ideal of $R$, $M$ is an $R$-module. 
$$IM\supset I^2M\supset I^3M\supset\cdots$$
What is $\mathop {\lim }\limits\_{\begin{subarray}{c} 
 \longrightarrow \\ 
\end{subarray}}I^nM$ ?



What is lim⟶ I^n M?
I'll interpret this question by agglomerating information given in the comments: We choose an element $x \in I$, and want to know the structure of the direct limit of $\{ I^nM\}\_{n \geq 0}$, when the maps $I^nM \to I^{n+1}M$ are given by multiplication by $x$.


I think the answer is that we can't say very much at all in general without somewhat more information, and the assumption that $x$ is $M$-regular is not enough. For example, if $R = \mathbf{Q}[t]$ and $M=R$, then choosing $x = t$ yields a direct limit that is the zero module, while choosing $x=1$ yields a direct limit that is $M$ (which is $IM$ in this case). For more general rings $R$, you can get intermediate results by setting $x$ to be something that has an interesting fixed subspace, e.g., an idempotent.


