I didn’t actually want to take that long of a break before this post, but I had to do a final exam and give/grade a final, so that ate up lots of time. The next natural thing to move on to is something called Krull dimension. This is sort of annoying to define, but highly useful. I’ve also decided I’m going to stop “fraking” my ’s, since it is annoying to type and just use capital P’s for prime ideals.

First we need to define something I’ll call “height.” A prime chain is a strictly decreasing chain of prime ideals: . Now we define the height of a prime ideal P, ht(P), to be the length of the longest prime chain with .

Some quick examples: It is easy to check that ht(P)=0 if and only if P is minimal, and hence in an integral domain ht(P)=0 if and only if . Let where k is a field. Then let be the prime ideal generated by those indeterminants (check that it is prime easily by noting which is clearly an integral domain). Then for any n, we can make a chain . Thus .

Now for the actual definition we want to work with. I’ll denote the Krull dimension simply by “dim” rather than “Krulldim”. Then we define: . So our quick example here is that for integral domains, if and only if is a field.

My goal for the day is to characterize all Noetherian rings of dim 0. The claim is that dim(R)=0 if and only if every finitely generated R-module M has a composition series. Since R is Noetherian, there are only finitely many minimal prime ideals. Since dim(R)=0, every prime ideal is minimal and hence there are only finitely many. Let’s call them .

Let’s look at the nilradical: . Since the radical is nilpotent, there is some m such that . So we define , so .

Let M be a finitely generated R-module. Then we have the chain . Now note that as a module is an -module. But is maximal and so a field, so it is a vector space. But M is finitely generated, so finite-dimensional, thus we can refine the chain so that all factors are simple.

Now we do this same trick on each of the chains : . Since at the m stage we get , we have a composition series for M.

For the converse suppose every finitely generated R-module has a composition series. Dimension zero is equivalent to showing that R has no prime ideals P, and Q such that . Suppose such exist. Let’s pass to the quotient, , and reinterpret our hypothesis. Then R is an integral domain that has a nonzero prime ideal and a composition series . So is minimal. Let be any nonzero element. Then since and (we’re in a domain), then by minimality we have . So there is a such that , i.e. . And hence . Thus R is a field which contradicts our having a nonzero prime ideal.

Well, I think that is enough fun for one day. I may post again tomorrow, since my final is Wed.

I realized there was one more result I probably should have included last time. Oh well. Here goes:

Let be integral, a prime ideal in R and prime ideals in lying over . If , then .

Proof: Recall that is integral over by last time for any multiplicative set, and also that prime ideals are preserved in rings of fractions. Thus the hypotheses still hold if we localize at . Thus is integral over , and are prime ideals. Thus we can WLOG replace and by their localizations and hence assume they are local. So now is a maximal ideal in . Thus by last time is maximal. Since , we have .

Now we are ready for the two big theorems. Here is the “Lying Over” Theorem. Let be an integral extension. If is a prime ideal in R, then there is a prime ideal in lying over , i.e. .

Proof: First note that and form the two sides of a commutative diagram. By last time is integral over . Choose a maximal ideal in . Thus is maximal in . But is local with unique max ideal , so . But the preimage of a prime ideal is prime, so is a prime ideal in .

Now we just diagram chase: . And also: .

Thus lies over .

Our other big theorem is the one about “Going Up”: If is an integral extension and are prime in R, and lies over , then there is a prime ideal lying over with .

Proof: By last time is an integral extension where is embedded in as . Now we just replace and R by these rings so that both and are . Now we just apply the Lying Over Theorem to get our result.

So as we see here integral extensions behave extremely nicely. These theorems guarantee that se always have prime ideals lying over ones in the lower field. This has some important applications to the Krull dimension that we’ll start looking at next time.

If you haven’t heard the terms in the title of this post, then you are probably bracing yourself for this to be some weird post on innuendos or something. Let’s first do some motivation (something I’m not often good at…remember that Jacobson radical series of posts? What is that even used for? Maybe at a later date we’ll return to such questions). We can do ring extensions just as we do field extensions, but they tend to be messier for obvious reasons. So we want some sort of property that will force an extension to be with respect to prime ideals. Two such properties are “lying over” and “going up.”

Let be a ring extension. Then we say it satisfies “lying over” if for every prime ideal in the base, there is a prime ideal in the extension such that . We say that satisfies “going up” if in the base ring are prime ideals, and lies over , then there is a prime ideal which lies over . (Remember that Spec is a contravariant functor).

Note that if we are lucky a whole bunch of posts of mine will finally be tied together and this was completely unplanned (spec, primality, localization, even *gasp* the Jacobson radical). First, let’s lay down a Lemma we will need:

Let be an integral extension of R. Then
i) If a prime ideal of R and lies over , then is integral over .
ii) If , then is integral over .

Proof: By the second iso theorem , so we can consider as a subring of . Take any element . By integrality there is an equation with the . Now just take everything to get that integral over . This yields part (i).

For part (ii), let , then , where and . By integrality again we have that , so we multiply through by in the ring of quotients to get . Thus is integral over .

I’ll do two quick results from here that will hopefully put us in a place to tackle the two big results of Cohen and Seidenberg next time.

First: If is an integral ring extension, then is a field if and only if is a field. If you want to prove this, there are no new techniques from what was done above, but you won’t explicitly use the above result, so I won’t go through it.

Second: If is an integral ring extension, ten if is a prime ideal in R and is a prime ideal lying over , then is maximal if and only if is maximal.

Proof: By part (i) of above, is integral over and so as a corollary to “First” we have one is a field if and only if the other is. This is precisely the statement that is maximal iff is maximal.

We now have the following equivalent definitions of the Jacobson radical. Remember right now we assume R is commutative with 1.

1) Intersection of all maximal ideals
2) Intersection of the annihilators of all simple left R-modules
3) The set of non-generators of R
4) The set of elements, x, such that has a left inverse for all r.

I think I already pointed out that from at least two of these definitions we automatically get that is a two-sided ideal. Two basic examples are now that if R is any field, then . And if K is a field, and , then . An important generalization is that in any local ring the Jacobson radical is the set of non-units.

An important result called Nakayama’s Lemma is that if is finitely generated, then implies that . Special case: If , then . Corollary to that special case: If , then $M=\{0\}$ (this last form is what is sometimes called Nakayama’s Lemma).

Proof: Suppose . Where , and for all j. Define .

Then with this setup, we exploit the non-generator definition. Note that



… etc
.

And we are done! It may have seemed a little roundabout to go through the “Frattini submodule” in developing the Jacobson radical, but it certainly pays off to have lots of definitions as we see here.

The last little bit I wanted to say was that we can define the Jacobson radical for a ring without identity. I don’t want to go through the details, but a standard trick is to define a new ring (with identity) with the standard addition, and then . It is pretty basic to check that where is some ideal in R (by the fact that ). It is also just algebraic manipulation to check that is the largest ideal in R such that for every there is a such that . This then is our definition. where is the collection of ideals satisfying that property.

First recall that we showed , and hence is a submodule of R as a module over itself. Thus is a left ideal of R. Next recall that we showed , and hence is a right ideal. i.e. is a two-sided ideal.

Let’s now work towards the annihilator definition. Define an equivalence relation the set of maximal ideals of R by ~ if there is a simple left R-module M with elements such that and . We see that this is an equivalence relation, since ~ iff and are isomorphic as R-modules. Examine the module homomorphisms and to see that if ~ then . Also, if by the iso , then , so ~ since with and .

Now let be an equivalence class of maximal left ideals. I claim that , where M is a simple left R-module isomorphic to each , for . By definition and the property above we get that . Thus if , then which means that where satisfies . But now this gives precisely .

Now just intersect over all the maximal left ideals. We get . And voila, we have it. This was a rather terse run-through and assumed a working knowledge of some facts about modules, but I find it to be a rather fascinating take on the development.

Next we’ll exploit some of these definitions to get some properties of the Jacobson radical, and develop it in a method that doesn’t require our ring to have 1.

Let’s think about what is going on in a different way. So now let’s think of elements of the ring as functions with domain . We define the value of the function at a point in our space to be the residue class in . This looks weird at first, since the image space depends on the point that you are evaluating the function.

Before worrying about that too much, let’s see if we can get this notion to match up with what we did yesterday. We have the nice property that if and only if . (Remember that even though we think of f as a function, it is really an element of the ring).

Define for any subset of the ring S the zero set: . Now from what I just noted in the previous paragraph, we get that these are just precisely the elements of that contain S, i.e. the closed sets of the Zariski topology. Thus we can define our basis for the Zariski topology to be the collection of .

We also will want what is “an inverse” to the zero set. We want the ideal that vanishes on a subset of Spec. So given , define . Now this isn’t really an inverse, but we get close in the following sense:

If is an ideal, then . Taking the ideal of the zero set is the radical of the ideal. And the radical has two equivalent definitions: .

If we take the ideal and zero set in the other order we get that : the closure in the Zariski topology.

We can abstract one step further and put a sheaf on . Note that for any we have that is a multiplicative set, so we can localize at it. Since I haven’t talked at all about sheaves, I’m not sure if I want to go any further with this, so maybe I’ll do some more examples next time and possibly start to scratch this surface.

So I’ve built up localization starting there, and I’ve built up the theory of prime ideals scattered throughout, but ending here. I also just assume the basics of topology in my posts, so we are in the perfect position to talk about a very fascinating construction and incredibly useful tool that combines all these things.

Warning: I have just started learning about this stuff, so it could be riddled with confusion or error. Luckily, I’m just posting the basics which some readers probably know like the back of their hand and will hopefully point out problems.

Of course what I’m referring to is Spec. As usual let’s assume that R is a commutative ring with 1 (I don’t think we need the 1). Then . So we have the collection of all (proper) prime ideals of the ring. Other than prime ideals being my favorite type of ideal, this seems to be useless right now.

Let’s put a topology on our set now (the “points” of our space are prime ideals). Let be any ideal. Define . Then we define the closed sets of the topology to be the family of all such sets, i.e. are the closed sets. This is known as the Zariski topology.

To check that these really satisfy the right axioms, (I won’t go through it, but) note that , , and (The last is probably the least trivial, but they all follow in a straightforward from definition way).

Examples:

1. If our ring is a field k, then the spectrum is a point.

2.Another common example would be . In other words, the prime ideals can just be identified with the prime number that generates them (and we have (0) as a special circumstance). So open sets are subsets of that are missing finitely many prime numbers. So we see that the Zariski topology is not Hausdorff (and rarely is). It will, however, always be compact.

3. Possibly the most important examples are the ones dealing with polynomial rings. In the nicest case, when k is an algebraically closed field, we have that since the prime ideals are just multiples of linear polynomials, we have the bijection of sending any to the prime ideal generated by (and we still have that pesky “zero” floating around that we’ll talk about later).

Last for today is that Spec is a contravariant functor from rings to topological spaces. We’ve basically done everything we need, since we see how it takes a ring object to a Top object. Also if we have a ring hom , then define in the obvious way, i.e. .

I promised some localization and we should be able to get to that next time, but there is just so much going on here that it is nearly impossible to exhaust (well, from my perspective as a newbie to the topic).

I promised this awhile back. It seems as if the Noetherian condition is really the last major thing I need before being able to move on.

A ring is Noetherian if every ascending chain of ideals stabilizes (or “terminates”). So, this means that given any collection of ideals such that we have that there exists some so that for all . This condition seems very strange at first. It is known as the Ascending Chain Condition, or ACC for short, but it turns out that it is equivalent to some other things and makes sure our rings are somewhat well-behaved.

Since for the purpose of this collection of posts we only care about commutative rings, the ACC is equivalent to the condition that every ideal is finitely generated.

Proof) Suppose every ideal is finitely generated. Then let be an ascending chain of ideals. Since is an ideal, it is generated by say m elements: . But each one of these elements come from some specific ideal, so suppose . Then just take and we have that the chain stabilizes after that.

For the reverse we go by contrapositive. Let be some ideal that is not finitely generated. Then we can find such that . We can also find such that . We can continue this process without termination. If it terminated at some step then, the ideal would be finitely generated. Thus we now just note that we have an ascending chain that doesn’t terminate .

It is easily seen that every PID is Noetherian. Rings tend to stay Noetherian under new constructions. The ring of polynomials (in finitely many indeterminates) and ring of power series where the coefficients come from a Noetherian ring is Noetherian. The former is known as the Hilbert Basis Theorem. Both the quotient and the ring of fractions are Noetherian if R is Noetherian.

But remember we want to figure out how this works with prime ideals. It turns out that prime isn’t quite what we want to get the best results, but in order to not introduce yet another type of ideal, I’ll leave this out since it won’t appear in anything I do later. So it turns out that if an ideal and R Noetherian, every prime ideal contains a minimal-over-I prime ideal, say . This is just a standard one step application of Zorn’s Lemma.

So I think I’ve beat primality to death. Next time I’ll do a sort of “history of math” type post on Hilbert’s Zahlbericht to put into the blog carnival. This will give me some time to think of where to go from here. I’m thinking the algebraic number theory side…I just don’t want to have to build Galois theory before I do it.

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.

Let’s figure out what “local” means and see if our construction somehow makes a local ring, i.e. is a “localization.”

Local: A ring is called local if there is a unique maximal ideal. This seems like a rather silly term, but it actually makes sense when you look at how rings arise in algebraic geometry or manifold theory. We won’t go there, though.

Sadly, it turns out that is not always a local ring. But this is where primality comes into play. If is a prime ideal then is a multiplicative set. Suppose it weren’t, then there would be two elements such that , i.e. , but this is impossible, since by definition either or . We now denote the localization of at , to be which we denote with the shorthand . This does turn out to be local since by the property listed last time of the embedding , so is the unique maximal ideal in .

Proof: Suppose , then with and . If , then is a unit in . So all nonunits are in . Now if I is any ideal in that contains an element with , then . Thus every proper ideal in is contained in . So is local with unique max ideal . For notational purposes outside of this blog, people usually write the prime ideal as and the unique maximal ideal of as .

I guess I’ve been rather sparse on the examples. The first one that comes to mind is surely to take . Then our prime ideals are just the principal ideals generated by the primes, so take for some prime p. Then .

I guess the importance of prime ideals leads us to explore some properties of prime ideals that could be useful.

Property 1: If is any multiplicative set (not containing 0) and if is a maximal ideal, then is prime. Also, any ideal is contained in such a . I’ll omit proving this. The first part is fiddling with things until it works and the second statement uses Zorn’s Lemma.

OK, well I thought I had some other properties, but I can’t seem to find them/think of them now. I’m not sure where I’m going next. I’ll either move on to some related things to get at this better like the nilradical, or I’ll generalize this one more time to modules and do it using the categorical construction. If anyone has suggestions on which of these paths to take, just post. You probably have a few days as I’ll get busy again.