Krull intersection theorem for Noetherian domains

Statement
Let $$R$$ be a Noetherian domain (i.e. a Noetherian ring that is also an integral domain) and $$I$$ a proper ideal in $$R$$. Then, we have:

$$\bigcap_{j=1}^\infty I^j = 0$$

Applying Krull intersection theorem for modules
The Krull intersection theorem states that if $$M$$ is a finitely generated module over a Noetherian ring $$R$$ and $$I$$ is an ideal inside $$R$$, then there exists $$r \in I$$ such that:

$$(1 - r)\left( \bigcap_{j=1}^\infty I^jM \right) = 0$$

We apply this to the case where $$M = R$$, to get that there exists $$r \in I$$, such that:

$$(1 - r)\left( \bigcap_{j=1}^\infty I^j\right)=0$$

Applying the integral domain condition and properness of the ideal
Since $$I$$ is a proper ideal, $$1 \notin I$$. Hence $$r \ne 1$$, so the element $$1 - r$$ cannot be zero.

Thus, by the fact that we are in an integral domain, and by the above equation, we get:

$$\bigcap_{j=1}^\infty I^j = 0$$