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Are there different ways to create prime ideals in a ring other than taking quotients? I recently came across a construction of a prime ideal in a Noetherian ring $A$ given in the book on Algebraic Number Theory by James Milne as follows. Take $c$ belonging to $A$, where $c$ is not zero and not a unit in $A$. Construct the $A$-module $M = A/(c)$ and choose an $m \neq 0$ in $M$ such that $Ann(m) \subset A$ is maximal (We can do this because A is Noetherian). This particular construction helped in proving a theorem about local, Noetherian and integrally closed rings (that they are PIDs). I wanted to know if you have come across any more such constructions.

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    $\begingroup$ The construction you mention is called "associated primes", and is one of the basic tools in commutative algebra. I suggest that you look at some introductory book in the subject (e.g. Atiyah-MacDonald). $\endgroup$
    – abx
    Mar 12 at 8:00
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    $\begingroup$ There are a few exercices in the second chapter of Eisenbud's book "Commutative algebra" that show that ideals that are maximal with respect to some properties are prime. $\endgroup$ Mar 12 at 8:10
  • $\begingroup$ I'm not sure if it's a construction, exactly, but minimal prime ideals have a similar flavor and are often important. $\endgroup$
    – Will Sawin
    Mar 12 at 23:40
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    $\begingroup$ There are many proofs in commutative algebra in which an ideal, maximal with respect to some property, is shown to be prime. For a general examination of this phenomenon, see Lam and Reyes, J. Algebra 319 (2008), no. 7, 3006--3027. $\endgroup$
    – user166831
    Mar 14 at 7:17
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I'll turn my comment into an answer. Indeed, there are many proofs in commutative algebra in which an ideal, maximal with respect to some property, is shown to be prime. "Of course the best known and probably most important is Krull's result that an ideal maximal with respect to missing a multiplicatively closed set is prime. Also, an ideal maximal with respect to not being principal, invertible, or finitely generated or an ideal maximal among annihilators of nonzero elements of a module is prime." In the paper listed below, the authors prove a metatheorem, the Prime Ideal Principle. They define the notion of a family of ideals $F$ being Oka or Ako. If $F$ is Oka or Ako, then any ideal maximal with respect to not being in $F$ is prime.

Lam, T. Y.; Reyes, Manuel L. A prime ideal principle in commutative algebra. J. Algebra 319 (2008), no. 7, 3006--3027.

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  • $\begingroup$ Wow, this seems quite interesting. Thank you for the answer, will go through this paper :) $\endgroup$
    – Yashi Jain
    Mar 17 at 7:15

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