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.

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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](https://mathoverflow.net/questions/109161/a-geometric-insight-into-a-proof-of-krull-intersection-theorem) 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?