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There is a related question regarding this, but almost all answers lean towards algebraic geometry, which I know none of. The wikipedia page says something related to algebraic number theory, another field I know nothing of.

So my question came up during reading Cohen's theorem which states

A ring $R$ is Noetherian iff every prime ideal of $R$ is finitely generated.

Basically this theorem says we can reduce the study to the prime ideals of $R$ instead of just "regular" ideals of $R$ since we know

A ring $R$ is Noetherian iff every ideal of $R$ is finitely generated.

So what is so fundamental in the role prime ideals play? And I don't mean in this particular case, but in general. Do they generate certain structure/objects in algebra like how primes generate all the other integers? Is there something "easier" with regards to prime ideals? Does reducing theorems to prime ideals give us something more fundamental?

Sorry if these questions sound all the same, I am just looking for some simple answers.

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    $\begingroup$ "So what is so fundamental in the role prime ideals play? " This sounds very broad. Prime ideals are connected to prime elements in a ring. For the ring of integers, prime numbers are very important, e.g., for prime number distributions (Riemann hypothesis). $\endgroup$
    – Dietrich Burde
    Sep 11, 2019 at 18:45
  • $\begingroup$ @DietrichBurde that actually triggered some ideas in my head. $\endgroup$
    – Lemon
    Sep 11, 2019 at 18:50
  • $\begingroup$ For one general viewpoint see Lam and Reyes: Oka and Ako Ideal Families in Commutative Rings, $\endgroup$
    – Bill Dubuque
    Sep 11, 2019 at 19:32
  • $\begingroup$ See also Noncommutative generalizations of theorems of Cohen and Kaplansky $\endgroup$
    – Bill Dubuque
    Sep 11, 2019 at 19:39
  • $\begingroup$ Note: the above two are generalizations of the "maximal implies prime" idea at the heart of Cohen's theorem. This is but one of many key roles played by prime ideals. $\endgroup$
    – Bill Dubuque
    Sep 11, 2019 at 19:52

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So what is so fundamental in the role prime ideals play? And I don't mean in this particular case, but in general.

This is well answered by the post you linked already. I don't really think "but I don't know anything about those branches so I don't want to hear reasons from those branches" is a valid reason to dismiss the other question. In dismissing them you're dismissing some of the nicest and most relevant cases.

It would be especially worthwhile to read up on the connection between prime ideals and irreducible varieties (for example, something like this).

Do they generate certain structure/objects in algebra like how primes generate all the other integers?

Well, the direct analogy for that is in Dedekind domains, yes. For those rings, ideals have prime factorizations, but this isn't true for rings in general.

Is there something "easier" with regards to prime ideals? Does reducing theorems to prime ideals give us something more fundamental?

Yes, that is trivially the case. If you re-characterize anything from terms of a large class to a smaller class, then you have "made things easier" in that you only have to check the small class instead of the large class. Reducing questions from the general case to constituents that make up all cases is a general theme.

The example you gave in particular about the relationship to Cohen's theorem was also covered in this earlier post.

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  • $\begingroup$ Regarding your first answer: I just don't have the time go through and survey the entire field(s) in one semester. I am studying for exams. Regarding your 3rd answer: Would the class of prime ideals be the "smallest (and most desirable) class" or are there "smaller" structures we can get? Is this the analogy to knowing about the prime ideals equates to knowing everything else about the big space? $\endgroup$
    – Lemon
    Sep 11, 2019 at 20:00
  • $\begingroup$ @Hawk You don't have to survey the entire field, I'm sure you can get the most important parts just by working on it for a day. Don't be deterred by such negativity! $\endgroup$
    – rschwieb
    Sep 11, 2019 at 20:01
  • $\begingroup$ I think probably the most useful subsets of the prime ideals are the minimal and maximal primes. Other than that, I haven't really heard of anything else that compares, with regards to usefulness or importance. $\endgroup$
    – rschwieb
    Sep 11, 2019 at 20:03
  • $\begingroup$ Would the pdf about varieties be sufficient? I am just suspecting many theorems later on in algebra will follow the pattern of reducing cases to its prime ideals. $\endgroup$
    – Lemon
    Sep 11, 2019 at 20:05
  • $\begingroup$ Yes, the last paragraph in my solution essentially says "knowing about the prime ideals equates to knowing everything else about the big space" or to paraphrase "knowing about the pieces tells you about the whole." Mind you, only some things are amenable to characterization with prime ideals. Some things have no connection with them. $\endgroup$
    – rschwieb
    Sep 11, 2019 at 20:05

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