Krull intersection theorem for modules

This fact is an application of the following pivotal fact/result/idea: Artin-Rees lemma
View other applications of Artin-Rees lemma OR Read a survey article on applying Artin-Rees lemma

This fact is an application of the following pivotal fact/result/idea: Nakayama's lemma
View other applications of Nakayama's lemma OR Read a survey article on applying Nakayama's lemma

Statement

Let $R$ be a Noetherian ring and $I$ be an ideal inside $R$.

• If $M$ is a finitely generated $R$-module, then there exists $r\in I$ such that:

$(1-r)(\bigcap _1^{\mathrm{\infty }}{I}^{j}M)=0$

• If $R$ is an integral domain or a local ring and $I$ is a proper ideal then:

$\bigcap _1^{\mathrm{\infty }}{I}^j=0$

Results used

• Artin-Rees lemma
• Nakayama's lemma