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?

I asked this question last year in MSE, but I didn't get an answer.

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

Are there any other simple or intuitive proofs for the theorem? I have 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 polynomials, this is easy to accept if we consider the polynomial as a Taylor series expansion (with finitely many terms). Using this, I tried to prove it for a local ring, as follows.

Let $R$ be a Noetherian ring. Suppose there exists $D:R\to R$ satisfying $D(ab)=(Da)b+a(Db)$ and $D(a+b)=Da+Db$, that is, it is a derivation on $R$. Define $\mathrm {Poly}(R):=\{r\in R\,| \,D^{k}r=0$ for some $k>0\}$, which means that the set of elements in $R$ on which $D$ behaves as a polynomial. 

We check that $\mathrm {Poly}(R)$ is a subring of $R$, and I wonder whether $R=\mathrm {Poly}(R)$. Then I want to approximate elements in $R$ via their Taylor expansions, but it was not easy to formalize this. I think that the evaluation map corresponds to quotients by maximal ideal; 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 rings (or Noetherian local rings). I also noticed that there aren't any nontrivial derivations on the ring $\mathbf{Z}$. Is there a proof that uses these ideas or versions of them?