System of parameters which have linear independent images in the cotangent space > Given a Noetherian, local ring $(R,m)$, can we always find a system of parameters whose images in the cotangent space $m/m^2$ are linearly independent? We can do this in the regular case, by just choosing a basis for the cotangent space and looking at their preimages in $m$ under the canonical map. By Nakayama's lemma these generate $m$ and hence form a system of parameters. Can we do this for any general Noetherian, local ring?

By induction let's just worry about picking one parameter element $x$. We need $x$ to be in $m$, but outside all the minimal primes of $R$ and $m^2$. This smells exactly like Prime Avoidance (note that they don't have to be all primes!). Now replace $R$ by $R/(x)$ and repeat.