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 if $R$ is a Noetherian ring and $M$ is a finitely generated module over $R$, and $I$ is an ideal in $R$, we have:

$I\left(\bigcap _{j=1}^{\infty }I^{j}M\right)=\bigcap _{j=1}^{\infty }I^{j}M$

We apply it to the case $M=R$. We thus get:

$I\left(\bigcap _{j=1}^{\infty }I^{j}\right)=\bigcap _{j=1}^{\infty }I^{j}$

Applying Nakayama's lemma

Consider the ideal $N=\bigcap _{j=1}^{\infty }I^{j}$ as a $R$-module. Since $IN=N$, and $I$ is contained in the Jacobson radical of $R$, Nakayama's lemma tells us that $N=0$. This is precisely what we want.