1

How do you motivate the concept of a ring-theoretic ideal, and subsequently prime/maximal ideals, to a first-time ring theory student who has never seen these things before?

Let's say they just know the definition of a ring, and some common examples --- that's it. There's two ways I can think of:

  • Use divisibility and posets: Say that divisibility in $\mathbb{Z}$ and $\mathbb{C}[x]$ corresponds to the poset of principal ideals, so you can use that to motivate the definition of ideals in general. The maximal elements in the poset are like the prime numbers. You could use the property "$p|ab\implies p|a \text{ or } p|b$" to motivate the element-wise condition of prime ideals, and all this motivates the question "when is every prime ideal maximal?" You can also use polynomials and irreducibles as an example, to even further motivate the need for generality.

  • Use the First Isomorphism Theorem: Motivate homomorphisms first and explain that ideals are the kernels of (surjective) homomorphisms. You could say that the cases of domains/fields recover prime/maximal ideals.

Or I guess you could mix-and-match ideas from both points.

What other motivations are good for undergraduates learning about ideals for the first time? Is there a way to get at $\mathrm{Spec}(R)$, even when the audience only knows what an ideal is?

Ehsaan
  • 3,227
  • I will probably get flamed by number theorists, but I think it's hard to motivate $\mathrm{Spec}(R)$ without referencing algebraic geometry. That's potentially different than motivating prime and maximal ideals, which is your main question, but I wanted to add that. For motivating "prime ideals", I think you just want to motivate ideals, and for that I think the historical context is really good. They were introduced ( I think by Dedekind ) in an attempt to salvage unique factorization in extensions of the integers, in an attempt to prove Fermat's Last Theorem. – Callus - Reinstate Monica Aug 05 '20 at 23:22
  • @Callus-ReinstateMonica why would you get flamed by number theorists for saying that? – KCd Aug 05 '20 at 23:31
  • Related wikipedia article: https://en.wikipedia.org/wiki/Ideal_number – Callus - Reinstate Monica Aug 05 '20 at 23:31
  • @Callus-ReinstateMonica Well, you could use the analogy between points $c\in K$ with linear/monic polynomials $x-c$, and then use the analogy between irreducible polynomials and prime numbers. Most students will buy that the poset of primes is probably similar to the poset of irreducible polynomials. You could mention how partial fractions works in both rational functions and rational numbers. So that could be a bridge to a bit of algebraic geometry ... – Ehsaan Aug 05 '20 at 23:31
  • If KCd stands for the K__ C__d number theorist I know, he certainly would not flame me for it. I wanted to be sure not to give the impression that (prime) ideals are only interesting from a geometric perspective. – Callus - Reinstate Monica Aug 05 '20 at 23:32
  • Taking the basic properties of the set of rational integers as the defining axioms of a commutative ring, we have as realizations of such structured sets the most dissimilar examples that have properties that the integers do not possess. For example, a local ring is defined as a ring having a single maximal ideal, which is not true in $\mathbb Z$ that has an infinity of such maximum ideals. Studying in detail very punctually can require a lot of elaboration. – Piquito Aug 05 '20 at 23:47
  • 1
    As far as I know, the original motivation for the introduction of ideals was to generalize the classical theory of divisibility of integers to situations, like rings of algebraic integers in number fields, where unique factorization can fail if stated in terms of elements of the rings but holds if stated in terms of ideals. In view of this purpose, prome ideals obviously came up very naturally as the analogs of the prime numbers in $\mathbb Z$. – Andreas Blass Aug 06 '20 at 00:05
  • @AndreasBlass I appreciate that and I think it's a good motivation. Do you know any others aside from the two I listed? – Ehsaan Aug 06 '20 at 00:28
  • There is a ton of good intuition spread out on the site, which I've tried to link. I have my doubts how well the spectrum can be explained to a beginner in ring theory, but if it can be done well then I think you'll find something here. – rschwieb Aug 06 '20 at 12:24

0 Answers0