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.^[1]

Statement of the lemma

Let ${\displaystyle I}$ be an ideal and let ${\displaystyle M}$ be a multiplicative system (i.e. ${\displaystyle M}$ is closed under multiplication) in a ring ${\displaystyle R}$, and suppose ${\displaystyle I\cap M=\varnothing }$. Then there exists a prime ideal ${\displaystyle P}$ satisfying ${\displaystyle I\subseteq P}$ and ${\displaystyle P\cap M=\varnothing }$.^[2]