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. $\square$

Comment #1066 by Thomas Smith on

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

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

Comment #1071 by Matthieu Romagny on

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

Comment #1074 by on

@Thomas: Try again with the complete slogan.

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

There are also:

• 7 comment(s) on Section 10.35: Jacobson rings

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).