Krull's separation lemma

In abstract algebra, Krull's separation lemma is a lemma in ring theory. It was proved by Wolfgang Krull in 1928.

Statement of the lemma

Let $I$ be an ideal and let $M$ be a multiplicative system (i.e. $M$ is closed under multiplication) in a ring $R$, and suppose $I\cap M=\varnothing$. Then there exists a prime ideal $P$ satisfying $I\subseteq P$ and $P\cap M=\varnothing$.