Lemma 10.35.17 (00G9)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 10: Commutative Algebra
Section 10.35: Jacobson rings
Lemma 10.35.17 (cite)

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Lemma 10.35.17. Let $R$ be a Jacobson ring. Let $I \subset R$ be an ideal. The ring $R/I$ is Jacobson and maximal ideals of $R/I$ correspond to maximal ideals of $R$ containing $I$.

Proof. The proof is the same as the proof of Lemma 10.35.14. $\square$

Comments (4)

Comment #1066 by Thomas Smith on October 09, 2014 at 00:50

Suggested slogan: The maximals which intersect to form primes are quotients of said primes by ideals of the primes.

Comment #1067 by Thomas Smith on October 09, 2014 at 00:58

Oops, should clarify that it is maximals of the quotient of the Jacobson ring by an ideal.

Comment #1071 by Matthieu Romagny on October 12, 2014 at 08:37

Just like for 00G6, I suggest: "correspond to maximal ideals of $R$ containing $I$ ".

Comment #1074 by Johan on October 15, 2014 at 12:54

@Thomas: Try again with the complete slogan.

@Matthieu: Thanks for this and for comment #1070.

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