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