Krull intersection theorem for Jacobson radical

Statement

Let $R$ be a Noetherian ring and $I$ an ideal contained inside the Jacobson radical of $R$. Then, we have:

$\bigcap _{j=1}^{\infty }I^{j}=0$

In particular, when $R$ is a local ring, then the above holds for any proper ideal $I$.

Proof

Applying the Krull intersection theorem for modules

We apply the Krull intersection theorem for modules, which states that