(Fix some local ring $(R,\mathfrak{m})$ over a field of zero characteristic.)

Suppose an ideal $J$ is defined by some complicated formula/procedure. And there is no hope of computing it/or writing down explicitly (I mean "analytically", not by computer).
But on the other hand the ideal has a geometric/algebraic meaning. So that some naturally related ideals are simpler. Then, instead of computing $J$ we could at least bound/approximate it.

For example: $\sqrt{J}\supseteq\overline{J}\supseteq J$. (the integral closure in the middle) The radical $\sqrt{J}$ can sometimes be computed "set-theoretically", by going over the points of $Spec(R)$. While for $\overline{J}$ one can use the criterion of projections onto DVR's (initially by Teissier). And over DVR things are usually simpler.
One can also try the saturation $J:\mathfrak{m}^\infty$


What are the other ideals naturally related to $J$ that are often "computable"?
(I'm interested primarily in various determinantal ideals, Fitting ideals, annihilator-of-cokernels etc. )