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