I've asked this question last year in MSE, but I didn't get any answer so I just want to post the question here, too.

I took a commutative algebra course last semester (with Kaplansky's book), and I've learned about Krull's intersection theorem. In the course, we proved it without using Artin-Rees Lemma. I heard that the standard proof uses the lemma.

By the way, is there any other simple or intuitive proofs for the theorem? I heard that there is an intuitive reason for coordinate rings in this post, that the only function which vanishes to arbitrarily high order at a point is the zero function. In case of polynomial, this is easy to accept if we consider the polynomial as Taylor series expansion (which ends in finite terms). With using this idea, I tried to prove it for a local ring :

Let $R$ be a Noetherian ring. Suppose there exists $D:R\to R$ satisfies $D(ab)=(Da)b+a(Db)$ and $D(a+b)=Da+Db$, i.e. it works as derivation on $R$. Define $Poly(R):=\{r\in R\,| \,D^{k}r=0$ for some $k>0\}$, which means the set of elements in $R$ works as polynomial. We can check that $Poly(R)$ is subring of $R$, and I wonder if $R=Poly(R)$ or not. Then I want to approximate elements in $R$ as a Taylor expansion and I want to prove the theorem, but it was not easy to formalize it. I think evaluation map corresponds to quotients through maximal ideal of $R$, and then it would be possible to find a function $r^{*}:k\to k$ where $k=R/\mathfrak{m}$ is a field and $r^{*}$ is induced by an element $r\in R$.

All these things are possible in the case of polynomial ring, but it is very hard to do it for general Noetherian ring (or Noetherian local ring). I also noticed that there isn't any nontrivial derivation on a ring $\mathbb{Z}$. Is there any proof that uses these ideas? Or how can I modify these ideas?