Point of Post: In this post we discuss the notion of maximal ideals, give several characterizations for commutative unital rings, etc.
Point of Post: In this post we define the notion of a prime ideal and for a ring , characterize prime ideals in commutative unital rings via their quotient rings, and find for a few specific rings.
We now come to what shall be one of the most important topics not only in the ring theory to come but in the application of ring theory to other subjects. Namely, in this post we shall discuss the notion of prime ideals. With a statement like that one would hope that there is some serious intuition to back it up. Thankfully, not only is the intuitive motivation for prime ideals abounding but it is accessible to us even with our current knowledge. In fact, there are two natural ways to motivate prime ideals, both with their own appeal. While either way is equally effective depending upon the goals of the readers, it is the bias of the author in saying that the first (we discuss) is ‘more important’. Whereas the second approach has the quality that it answers a ‘natural’ question the second answers a ‘deep’ question. Anyways, enough of the pre-preamble:
The first motivation for prime ideals is the one suggested from their name. In particular, I have said before that in a lot of ways the integers provide a guiding example for a lot of the ring-theoretic concepts that we shall discuss in these beginning posts. To this end it does us some serious good to examine the concepts of which proved most fruitful and see how precisely we may generalize these to fit our ring-theoretic purposes. Well, doubtlessly obvious to anyone who has done any kind of basic number theory (in fact, to anyone who has done any kind of mathematics) that one of the most striking and deeply important objects in are the primes. In particular, the fact that every integer may be written, up to units, uniquely as a product of primes. It would be fantastic if we could find a more general sort of ring for which this unique factorization into is possible. The first step in this is the necessity to define what precisely prime should mean in a general ring. While there are many equivalent ways of defining primes in , after some experimentation though one discovers that the key characterization of primes is Euclid’s lemma. Said differently, the primes of are precisely the non-unit numbers with the property that whenever one has or . That said, since most of our focus has been on ideals it would be nice to phrase this in such a language. In particular, it’s easy to see that a number is prime if the generated ideal is proper (i.e. not equal to ) and has the property that implies or . Or, taking it one step further the equivalent definition that and (recalling the product of ideals) implies or . Thus, we are lead to the notion of prime ideals in a general ring which are ideals with the property that implies or . Therefore, we are led, intuitively for now, to define a prime element to be one such that is prime. Moreover, we shall see that prime ideals shall take the place of primes in a very real sense which shall be made clear later.
For the sake of convenience we assume for this second motivational point that we are dealing with a commutative unital ring . We have seen that given an ideal we can form the quotient ring . While the existence of this ring is all nice-and-well (especially in consideration of the first isomorphism theorem) it’s not immediate what properties one can expect to have besides the ones it inherits being a homomorphic image of . For example, we may wish to know when is an integral domain. Exploring this further we see this is the same as asking when implies or . Of course, this is equivalent to having that implies or . Thus, also noting the fact that if and only if we see that for commutative unital ring one has that the prime ideals of are precisely those for which is an integral domain.