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I'm currently studying ring theory in my algebra course and I must admit I'm kind of lost. I was given the following problem:

Let S be a multiplicative system of R, a ring. Prove that the set $f=\{ I\triangleleft R\mid I\cap S=\emptyset\}$ has a maximal element and that $f$ is a prime ideal.

Since a multiplicative system is defined as a subset of R with no zero divisors, with the identity element included and such that for every $x,y\in S\implies xy\in S$, I can assure R has an identity. Furthermore, I can assume it is conmutative, since that is the kind of rings we are studying.

For the first part of the excercise I know I have to use Zorn's Lemma, but honestly I don't truly understand how to do that. How can I prove it is partially ordered? And how can I find an upper bound for every chain?

Thanks in advance, and please make me notice any mistake in my formulation. I'm new in the forum and I'm not quite sure how to do it.

JorgeOvi
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  • Well, the word "maximal" doesn't make sense without a partial order, so it had better be partially ordered if you're going to find a maximal element! If you understand what "maximal" means in this context, you must be at least implicitly aware what the partial order is. – Jack M Dec 12 '19 at 20:10
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    The collection of ideals disjoint from $I$ is partially ordered under inclusion. You need to show any chain of such ideals has an upper bound... – Angina Seng Dec 12 '19 at 20:12
  • Okey, so is that the union of these ideals? – JorgeOvi Dec 12 '19 at 20:16
  • See Theorem 1 in my answer in the linked dupe. For details of Zorn on ideal chains e.g. see here. You can find it in many places here e.g. search on "Zorn ideal chain union". If you're still stuck after all that then say where in comments here. This is a FAQ. Please search before asking questions. – Bill Dubuque Dec 12 '19 at 20:25
  • Just went through your answers, doubt solved (I actually understood the problem was not even properly formulated). I've seen lots of examples on how to use Zorn's lemma, but really struggle to apply it myself in new cases. I guess I will have to work on that. – JorgeOvi Dec 12 '19 at 21:04
  • Glad to see you've cleared it up. I think it takes time for everyone to becoem proficient with maximal principles like Zorn when they first encounter them. The real world doesn't lend any intuition on such matters so you have to start from scratch. – Bill Dubuque Dec 12 '19 at 21:22

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