Why is the Artin-Rees lemma used here?

Solution 1:

There is a proof due to Herstein that is "elementary" in the sense of avoiding the Artin-Rees lemma. Below is Kaplansky's presentation of this proof, from his "Commutative Rings". A proof using primary decomposition (as in Krull's original proof) can be found in Zariski and Samuel.

Solution 2:

See Theorem 3.6 in Chapter 6 of the CRing project for another elementary argument due to Perdry (American Math. Monthly, 2004) using only the Hilbert basis theorem. In fact, it shows more: if $R$ is a noetherian domain, $I \subset R$ a proper ideal, then the intersection of the powers of $I$ is trivial.