## Prime Ideals (Pt. I)

**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.

*Motivation*

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.

*Prime Ideals*

In accordance with the above definitions we define a *prime ideal *in a ring to be an ideal with the property that if are such that then or . We call the set of all prime ideals of a ring the *spectrum of *and denote it .

For commutative rings this definition simplifies considerably, but first some notation. We call a subset *multiplicative *if it is non-empty and closed under multiplication. What we now claim is that if is commutative then a proper ideal is prime if and only if . In particular, if is furthermore unital then is a prime ideal if and only if is a submonoid of . Indeed:

**Lemma: ***Let be a commutative ring, then for any one has that .*

**Proof: **Evidently since . Conversely, every element of is of the form

with and . But, then we have that from where the reverse inclusion and thus the lemma follows.

**Theorem: ***Let be a commutative ring. Then, if and only if is multiplicative.*

**Proof: **Suppose first that is multiplicative and let be such that . Now, if neither or then we could find and and so by assumption we have that but since this is a contradiction.

Conversely, suppose that is prime and . If we knew that this implied we’d be done since this would imply either or . But, this is precisely what the lemma preceeding this theorem says. The conclusion follows.

With this we are able to formally relate our two motivations by the following important theorem whose proof is really what we discussed in the motivation):

**Theorem: ***Let be a commutative unital ring. Then, if and only if is an integral domain.*

**Proof: **Suppose first that then we have, from basic discussions of quotient rings, that is a commutative unital ring with (this follows since ). Thus, it suffices to prove that has no zero divisors. To do this we merely note that if then by definition we must have that and so by the previous theorem this is equivalent to or which is equivalent to or . Since were arbitrary the fact that is an integral domain follows.

Conversely, suppose that then since in we must have that and moreover we see that since has no zero divisors then implies and so by definition and so by definition . Since were arbitrary we may conclude that is multiplicative from where the conclusion follows.

This theorem turns out to be a double-edged sword. On one hand, we have a theoretical factory for churning out integral domains, namely just finding commutative unital rings and prime ideals of such rings and modding out. On the other hand, this theorem takes the sometimes formidable problem of proving an ideal is or is not prime and reducing it to checking a property of its quotient ring. But, why is this easier? Well, there are two reasons really–firstly modding out often ‘takes away clutter’ in the sense that a lot of unimportant stuff ‘goes away’, secondly in the most natural cases we are provided with a nice description of the quotient ring via the first isomorphism theorem (e.g. the fact that ).

**References:**

1. Dummit, David Steven., and Richard M. Foote. *Abstract Algebra*. Hoboken, NJ: Wiley, 2004. Print.

2. Bhattacharya, P. B., S. K. Jain, and S. R. Nagpaul. *Basic Abstract Algebra*. Cambridge [Cambridgeshire: Cambridge UP, 1986. Print.

3. Reid, Miles. *Undergraduate Commutative Algebra*. Cambridge: Cambridge UP, 1995. Print.

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