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 $I$ 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 $P\cap M=\varnothing$ .^[2]

References

^ Krull, Wolfgang (1928). "Zur Theorie der zweiseitigen Ideale in nichtkommutativen Bereichen". Mathematische Zeitschrift. 28 (1): 481–503. doi:10.1007/BF01181179. ISSN 0025-5874. S2CID 122870138.
^ Sun, Shu-Hao (1992). "On separation lemmas". Journal of Pure and Applied Algebra. 78 (3): 301–310. doi:10.1016/0022-4049(92)90112-S.

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[Krull1928-1] Krull, Wolfgang (1928). "Zur Theorie der zweiseitigen Ideale in nichtkommutativen Bereichen". Mathematische Zeitschrift. 28 (1): 481–503. doi:10.1007/BF01181179. ISSN 0025-5874. S2CID 122870138.

[2] Sun, Shu-Hao (1992). "On separation lemmas". Journal of Pure and Applied Algebra. 78 (3): 301–310. doi:10.1016/0022-4049(92)90112-S.

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