Let $R$ be a local Noetherian ring over a field, with the maximal ideal $\mathfrak{m}$. (e.g. $R=k[[x_1,\dots,x_{p>1}]]$) Given two $R$-modules, $N\subset M$, of the same (finite, non-zero) rank. Their quotient, $M/N$, is torsion. I guess not much can be said in general.
Consider the quotients $(\mathfrak{m}^k\cdot M)/(\mathfrak{m}^k\cdot N)$. For $k\gg1$ this can be thought as some sort of saturation of the initial quotient.
If $M,N$ are ideals in $R$ then in some examples this sequence 'stabilizes' and 'simplifies' in some sense. For example:
1. The Loewy length of $(\mathfrak{m}^k\cdot M)/(\mathfrak{m}^k\cdot N)$ seems to stabilize.
2. If $M$ is free then the Loewy length of $(\mathfrak{m}^k\cdot M)/(\mathfrak{m}^k\cdot N)$ seems to non-increase with $k$.
3. $(\overline{\mathfrak{m}^k\cdot M})/(\mathfrak{m}^k\cdot M)$ seems to converge to the zero module. (Here the denominator is the integral closure.)
Are these properties true/well-known? Any other properties showing that $(\mathfrak{m}^k\cdot M)/(\mathfrak{m}^k\cdot N)$ is 'simpler' than $M/N$ for $k\gg1$? Which properties remain true if one replaces $\mathfrak{m}$ by an arbitrary ideal?
