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 $$P \cap M = \varnothing$$.