I want to wrap up some loose ends on the greatness of prime ideals before moving on in the localization theme. So. Recall that we formed the ring of quotients just like you would form the field of quotients. Only this time your “denominator” can be an arbitrary multiplicative set and this construction only gets us a ring. Moreover, this ring is not necessarily local. If we do the construction on a ring R with and the multiplicative set where P is a prime ideal, then we do get a local ring and we call this the localization.
Definition. Unique Factorization Domain (UFD): An integral domain in which every non-zero non-unit element can be written as a product of primes. (Note that there are equivalent definitions other than this one).
Quick property: Every irreducible element is prime.
Thus, it is instructive to look at some properties of prime ideals. First off, let’s look at the special case of UFD’s. It turns out that if R is a UFD, then for a multiplicative set S, is also a UFD. This mostly has to do with the fact that is an embedding and anything in is associate to something in . This makes a nice little exercise for the reader.
So what’s so special about prime ideals in UFD’s? Well every nonzero prime ideal contains a prime element.
Proof: Suppose and P prime. Then there exists , such that where a unit and prime. Thus . But this means that and since it is prime we have some .
Theorem: If R is not a PID, and P an ideal which is maximal with respect to the property of not being principal, then P is prime (and will always exist).
Sketch of existence: Zorn’s Lemma. The proof of this contains lots of nitty gritty element-wise computation and a weird trick, so I don’t see it as beneficial. What is beneficial is that we get this great corollary: A UFD is a PID if and only if every nonzero prime ideal is maximal.
I’ve been kind of stingy on the examples, so I’ll leave you with a pretty common example of a ring of fractions. These are usually called dyadic rational numbers. Take your ring to be . Then take your multiplicative set to be . Now are just the rational numbers with denominator a power of 2.
More generally we can form the p-adic integers (although that term is laden with many meanings, so I hesitate to actually use it). Let . Where we have the restriction iff satisfies . So elements of the ring are sequences. (Note embeds naturally since satisfies that relation). This is a ring with no zero divisors, so we can take it to be the multiplicative set and we get the field of fractions . The multiplicative group has a nice breakdown as .
Next time: Why Noetherian is important. How primality relates to it. And possibly another example.