Categories? Rings?

Well, things can get ultra busy around mid-terms. I don’t think I’ve posted in two weeks. What I really wanted to do next was to post some category theory basics. I’m not sure if I should, though, since so many math blogs have already done this. I then wanted to go on to define the fundamental group purely in categorical language. It turns out to be a really nice construction compared to the tedious typical way.

Instead, I’ve recently become quite interested in rings. There also seems to be a very large lack of “pure” ring theory in the blogosphere. Sure rings pop up and are needed by people doing things with algebraic geometry, say, but using ring theory isn’t the same as developing it.

I’m going to cover the basics quite quickly with the assumption of previous exposure, since I really want to get to some of the more interesting constructions (i.e. localization), then I’ll slow it down.

Ring: We have a set with two operations, we’ll call them addition and multiplication. The addition part forms an abelian group, and the multiplication…well, it puts you back in the set and is associative. We need a way to relate these operations, so we also require that a(x+y)=ax+ay and (x+y)b=xb+yb, i.e. there is a distributive law in effect. Note that there is not required in general to be a multiplicative identity, multiplicative inverses, or commuting of the multiplication.

NOTE: Until I say otherwise I will assume the ring is commutative (meaning multiplication) with 1 (meaning having a mult identity). R will always denote this.

Subring: A subset of R that is itself a ring.

Ideal: A subring that “swallows” multiplication. So I\subset R is an ideal if for any a\in I we have that ra\in I for all r\in R.

Prime ideal: An ideal I is prime if for element ab\in I we have that either a\in I or b\in I.

Principal ideal: An ideal I is principal if it is generated by a single element. So an ideal is generated by “a” if I=Ra=\{ra: r\in R\}.

Maximal ideal: A proper ideal I such that there is no other ideal K with the property I\subset K\subset R (where all containments are proper).

Domain: A ring in which the cancellation law holds. i.e. if ab=ac and a\neq 0, then b=c. Note that no element can divide zero, so if ab=0, then either a=0 or b=0.

We can quotient in the natural way: R/I is the set of cosets of I where our operations are (a+I)+(b+I)=(a+b)+I and (a+I)(b+I)=ab+I. We get the nice result that any ideal of R/I is of the form K/I where K is an ideal of R (and I\subset K\subset R).

I think that may lay down all the terminology I’ll need to get started. I’m not sure if I’ll really use any of these terms for awhile, though.

Common rings: \mathbb{Z}, \mathbb{Q}, \mathbb{R}, \mathbb{C}. Note that we can get \mathbb{Q} from \mathbb{Z} by taking “quotients.” This can be made precise for any domain. It is called the fraction field of R denoted Frac(R).

Let me take some time to explain this, since it is the motivation for localization. We want to form a field F that contains R as a subring such that the elements of F, say f\in F have the form f=ab^{-1} where b\neq 0. Note that this “looks” like division, and in fact is division in the case of \mathbb{Z}.

To make this process precise takes a bit of work, though. Set up X=\{(a,b)\in R\times R : b\neq 0\}. Define (a,b)~ (c,d) iff ad=bc. This is done since we want our relation to look like fractions, a/b , so a/b=c/d if we can cross-multiply and get the same thing. It is straightforward to check that this defines an equivalence relation. Now we let F be the set of equivalence classes.

Our operations on F should mimic those of fractions, so our addition is [a,b]+[c,d]=[ad+bc, bd] and [a,b][c,d]=[ac,bd]. These are well-defined and it is just computation to check that the axioms of a field are satisfied. (If you want a hint: the zero is [0,1] and the 1 is [1,1], the additive inverse of [a,b] is [-a,b] and the mult inverse is [b,a]).

Before finishing up, I want to point out how restrictive we had to be. We want a more general way of doing this. We don’t want to require that R be a domain, and we don’t want to have to take fractions with every single element in R. It turns out this general process is extremely useful and it is called localization.


Issues with Stewart

The old faithful Calculus text of Stewart has provided me with countless headaches this quarter. I must confess that my calc class is really quite bright. These problems do not occur in your standard freshmen calc class.

The Catch-22:
Stewart presents the definition of continuity at a point. A function is not continuous at “a” if it is not defined at “a.” Got that? Now he goes on to say a function is “continuous” if it is continuous at every point in its domain. This means that the function f(x)=\begin{cases} 1 \ \text{if} \ x>0 \\ -1 \ \text{if} \ x<0\end{cases} is continuous. Fine and dandy. Now the bright student pipes up and asks, how can it be continuous since we know it isn’t continuous at 0? Excellent question! A function can be continuous even if it isn’t continuous at every point. Wait! What? You just said that was the definition. No, no. I said a function is continuous if it is continuous at every point in its domain.

Etc. Etc. This has been the source of questions nearly everyday since this was covered. In fact, now that we are well into differentiation and things like “a function that is differentiable is continuous” has provided a new source of attempting to get the students to keep all the examples straight of differentiable not continuous, not continuous, not in the domain, is a function continuous at a vertical asymptote, etc.

My fix for the problem is quite simple. Let’s go slightly more general and define continuous on a set. This sounds horrific from the veteran professor’s point of view that can’t even get the students to understand continuous at a point or on its domain, but hear me out. If we start there, then have them do drill exercises such as is f continuous on (1,2) what about [1,2], what about R, what about etc. Then this concept of thinking in terms of sets is not really any more abstract than what they already do. The drill exercises will make sure they understand the concept. Now they can note that “at a point” really means, on the set {a}. Also, we now just specify continuous on its domain, and they realize that this is also a set. I’m not saying we should give them the Dirichlet function and ask whether it is continuous on the rationals or something. But intervals and unions of intervals are things they can do.

Why this fixes the problem: the term “continuous” will no longer be used in an ambiguous sense. Right now the source of confusion seems to come solely from the fact that it hasn’t been drilled into them that secretly you can’t say the word continuous without meaning “on a set.”

The catch-22: As a textbook writer, Stewart most certainly wanted to adopt a convention that would mean he could drop the phrase “on its domain,” (which I concede is explicitly stated, but what freshmen reads their text). It would be rather annoying to both read and write that phrase all the time, especially when it is so natural a convention (what would one even mean by saying continuous and including points that aren’t even in the domain of f?!?). There are also many students struggling with the concrete side of things, so introducing “continuous on a given set” is bound to completely lose those students. We should definitely consider the flip-side, though. This text is meant for a freshmen calc class, and so if drilling this concept and explicitly writing out something that isn’t very clear (to them) is going to help in understanding, then I think it is worth it.

Any fellow calc teachers think this is a good/bad idea? I typed this in a heated frenzy and have been thinking about it for quite some time, so it might not be crystal clear what my complaint is.

Land of Talk

I find this pattern rather interesting. What is the best album of the year so far? Of course, it is Bon Iver’s For Emma, Forever Ago. It is a hands down win for me. The new Land of Talk album Some are Lakes is really fantastic. It is quite genre defying. Often times your ear tells you that a song is in well-defined key. The problem is that there is so much dissonance and chaotic melody that it is an oversimplification to say that it is “really” in a key. At no point does your ear yell at you, though. It sounds perfectly natural. Like I said, it is amazingly original in sound.

The lyrics are often very blunt and challenging. I love the no-nonsense of them. There is no fluff here. Some examples:

We’ve seen how sick winds blow,
But i’ve got your boat for a night.
And I’ll love you like I love you then I’ll die.


One way road
Don’t care what I find
A little thunder’s good,
I thought maybe you would
But it’s okay,
We all feel left out
Sometimes growing up,
it can get you down.

So here is where the interesting part comes in. This is the only album that I think can rival Bon Iver. But Bon Iver is the name of Justin Vernon’s project. Who produced Land of Talk’s album? That’s right. Justin Vernon. It almost seems unfair that he could knock his own album off of the number one spot by producing another band’s album. At the same time it is fascinating that everything he touches can be so good. I wouldn’t be surprised if many prominant top 10 or 50 lists (KEXP or Pitchfork) considered both of these in the top 5.

Ethical Voting Habits

Is it ethical to vote based on well-informed opinions? By well-informed opinions I mean an opinion in which an educated rational person could successfully argue both sides. Take abortion for instance. I have an opinion on it, but I can also put that opinion aside and “successfully” argue either side. This is in opposition with something like an ill-informed opinion in which no rational person could successfully argue for slavery, say. If you are an issue voter voting on ill-informed opinions, then without discussion I’ll consider that unethical or at least ignorant and irresponsible.

The idea now is that we should be free to vote based on our (well-informed) opinions. The side with the most number of people wins and policies are created based on the majority. But what happens if fact comes up that doesn’t allow us to vote based on our opinions? This is our current situation. It has become apparent that Palin is not fit for the VP position. These are indisputable facts. She cannot name a single major newspaper of the world. She has no experience on a global scale. She believes things that are not true (e.g. the world is 6000 years old). When asked a question she has not been told the answer to, she becomes completely incoherent or changes the question to one that she has been told the answer to.

We have been put into a moral dilemma. Are we ethically obligated to put our opinions aside and vote for the qualified side? I hate to apply to a utilitarian argument, but it is very overwhelming at this point. We are not some small country that can do what we want (i.e. vote on single issues or on our opinions alone). We must take into consideration the rest of the world. We are voting into office a world leader, yet the rest of the world doesn’t get to vote.

Suppose for the next four years people continue to have abortions (insert issue of choice). The world continues. Suppose McCain is elected and then dies a few months or even years in. I’m not saying the world will end, but it will be drastically changed and not in a good way. How can Palin have reasonable conversations with other world leaders when at this point they probably don’t respect her? She won’t be taken seriously and her presence at these meetings will be purely for show.

Outside of America, abortion (insert any issue) isn’t really even considered an issue. To ignore the rest of the world in this decision and vote for a world leader based on something the rest of the world is not concerned about is to vote unethically. Thus, the ethical decision when fact turns up, is to put aside your opinions and vote in alignment with fact.

Clearly McCain’s choosing Palin was the initial unethical decision, since assuming a rational and ethical population, we no longer have the freedom to vote on our opinions. Now that his unethical decision was made, we must shed our opinions and partisanship and vote against him.