I am not a fan of comprehensive final exams. Well, that is not entirely true. I am not a fan of
taking comprehensive final exams. From the standpoint of a (potential) educator, I am an ardent supporter of comprehensive final exams. To me, perhaps the greatest proportion of learning and perhaps also the longest lived and therefore most useful is the reflective learning that can (should) happen at the end of a term. It is then that the subject's basic vocabulary is finally falling into place for the student, then that fundamental concepts are revealing their true forms and then that the long threads and strong connections are able to shine.
It is not in the power rule that the nature of the calculus lies, not even in the fundamental theorem but in the thinking approach, the notion of limit, convergence and divergence. It is no wonder advanced calculus courses (and textbooks) spend so much time defining limits and convergences and so (comparatively) little time on derivatives and integrals (as if these were just applications of limits). But not to harp on mathematics curricula, I actually think mathematics has a distinct advantage over other disciplines when it comes to comprehensive exams--the axiomatic nature of math demands this be true.
The problem with comprehensive final exams, I think, is not in the exams themselves at all (though I've sat through some ridiculous final exams--here's looking at you Design and Analysis of Control Systems) but instead in the unfortunate logistical constraint they must impose. By definition, a final exam must come at or near the completion of a course. And this period is also the busiest of all times in the term. Reading weeks and dead weeks help this some but the pressure of looming exams in multiple subjects doesn't make for much quiet reflection. It's almost like the last week (or more) of a term should be set aside for a retracing of our steps, a look back at the journey of the term with all the benefits of hindsight and sophistication of (hopefully) education. At this point, we are not concerned with homeworks and small quizzes. At this point we have come full circle and are again interested in concepts and connections (and where appropriate or interesting, we are now able to dig into the details and appreciate the nuances).
I would like to do that. Perhaps if I teach a course under my own power and authority someday, I will do that. But for now, through my own doing and the doing of others, I have about 18hrs to reflect and prepare for my comprehensive final exam (in Abstract Algebra II--Ring Theory and a dab of Field Theory). So, I'd like to take a shot. I'd like to tell the story of my/our journey through the semester(s). We're talking about concepts so I'm not going to do a lot of proofs here or even give full definitions. Here goes. And here's to hoping that my theory pans out and that this is actually a good use of study time and not just me wasting time on Xanga (a distinct possibility).
Ring Theory
MOTIVATION
We start with the integers. The integers are a hugely useful concept--all of number theory is based on them. Yet there are many things in this world that are not integers--5 apples for example is not the same as 5 the number. But there is something distinctly 5 about 5 apples. The integers are an abstraction of our world. What is amazing is that 5 apples has properties that are true for 5 oranges and even 5 Boeing 777's. 5 means something and the integers have properties that can be applied to other things.
And in the same way, we get the feeling that there are things "like" the integers elsewhere too--looking at an old-style clock for example where we go from 12 O'clock to 1 O'clock instead of 13 O'clock or in binary numbers going from 001 to 010 instead of 002. We would like to begin to capture some of the properties of integer-like things and so we define a mathematical object that bears them and try to see what we can do with it.
RING
That object is a RING. It consists of a set of things (could be integers, could not be) and it has two operations associated with it. We can "add" things in it and we can "multiply" things in it. "Add" and "Multiply" need not look anything like what add and multiply look like in the integers. Add means whatever add means for the set of things and likewise for multiply. Of course add and multiply have to follow certain rules. Addition must be commutative and associative (like in the integers) and there must be things like negatives in the set to allow us to cancel things with addition. Multiplication must be associative and follow the distributive property. Notice that multiplication doesn't have to be commutative and that it doesn't have to have "reciprocals" so we can't necessarily just cancel things in multiplication.
Surprisingly (wink wink) the integers form a ring. So do the real numbers and so do the complex numbers. In fact so do the integers mod n (like with the clock--that's integers mod 12. Binary is integers mod 2, so this stuff already is kinda useful--for telling time and computers and things like that.). We can see already that there are things (called SUBRINGS) that can living inside a ring but act like a ring themselves--like the real numbers living in the complex numbers or the integers living in the real numbers--these are subrings. The operations are the same but the elements are restricted to a "smaller" (in an infinite sort of way) set.
COMMUTATIVE RINGS AND THE BIG SPLIT
Now that we've created this ring object and come up with a few examples of rings, we can start to classify different kinds of rings. First off, we make a big split between rings where multiplication is commutative and rings where multiplication is not commutative. The integers are a ring where multiplication is commutative. The set of non-singular 2x2 matrices is an example of the non-commutative kind (There are 2x2 matrices where A*B /= B*A and they're not as uncommon as one might hope.). As you might expect, non-commutative rings are somewhat messier than the commutative kind (they're less integer-like after all). The vast majority of the rest of the term is spent on commutative rings. We shouldn't forget about non-commutative rings (not the least of which because matrices do a lot of things in our world) but we're going to put them on the side for now.
INTEGRAL DOMAINS AND FIELDS
The first commutative ring we look at is the INTEGRAL DOMAIN. This ring has the handy feature that you can cancel things with multiplication. So if ab = cb, a = c (just like in the integers--guess what the integers are integral domains). This isn't as impressive as saying "there's an inverse for every element in multiplication" which is like saying for every b there's a 1/b so b*1/b = 1. We save that property for later. But it does give us an important property. Integral domains present a smaller big split. We'll spend the rest of the term mostly assuming our rings are at least integral domains. (As a side note our integers mod 12 example is not an integral domain. It turns out you need to be mod p where p is a prime number.)
I say "at least" because some integral domains are special. They have multiplicative inverses--the b*1/b = 1 property. We call these rings (integral domains) FIELDS. So not only do fields have commutative properties and cancellation properties, they have cancellation properties in this special inverses-way. Here's where we part with the integers. Why? Who's the inverse of 2? 1/2? Nope. 1/2 is not an integer. Gotcha. But the reals are a field and so are the complexes. And our integers mod p where p is prime qualifies too. So that gives our basic vocab of rings.
IDEALS AND FACTORS
Now we split from the integers even more. We take a ring and define a special kind of subring--one that absorbs all the other elements of the ring by multiplication (It sounds bad but it's really good.). It means if there's an element a in the ideal subring, and another element r in the big ring but not in the subring, the product a*r actually lives in the subring as well as the big ring.
Why do we care? I struggle with this one. It's hard to see the intrinsic value of ideals (maybe because they're not like things in the integers that we normally deal with. We'll here's a shot. In the integers, ideals turn out to be things like the even numbers, the multiples of 3, multiples of 4, multiples of any integer. You can multiply one of those elements by any other element and still get a multiple of 3 or 4 or 39, etc. The thing that is important about these ideals is the break the big ring up into distinct partitions (if we have the multiples of 3, the rest of the elements are either multiples-of-three + 1 or multiples-of-three + 2 (where the multiples-of-three are really multiples-of-three + 0). We call the sets that make up these partitions COSETS (there's some French or German translation that makes this make more sense I'm sure, some clever mnemonic that's lost on us today.)
Well, why do we care? hehe. Well, if you imagine taking the integers and then dividing them (all of them) by the set of all multiples of three you get a set of new elements, cosets of multiples-of-three and it turns out that these cosets themselves form a ring (it's called a FACTOR RING). It has three elements 0, 1 and 2 (in a manner of speaking) because you're always either 0, 1 or 2 away from a multiple of 3. Well, this new ring is really an old ring. It's integers mod 3 which is great because we know a lot about that ring already--for one thing it's a field.
And it turns out that since it's a field, multiples-of-three is the "largest" or maximal ideal in the ring that contains multiples of 3. This may not sound too impressive but think about multiples-of-six. They're all completely contained in multiples-of-three and then some. So multiples-of-three is a "bigger" ideal. To say nothing of multiples-of-nine or multiples-of-81... Can you guess what's next? Theorem: All the maximal ideals of the integers are ideals that are multiples-of-primes (since there are no factors of other numbers hiding inside the primes that could make a "bigger" ideal). And so all factor rings created from dividing out multiples-of-primes are integers mod primes and therefore are fields.
Ack!! Ok, that's the first 2 chapters of the semester. I think we're doing ok (there's only 3 more to go, but it gets harder). But it's also 10:21pm now and I'm running out of time. I'm going to stop here and go study things at a slightly faster pace. Hopefully when all is said and I am done, I'll come back and finish up (or at least do a better job of quitting). After all, I've got the Two Towers of Rings (probably about homomorphims I think) and the Return of the Ring (sounds like a good place to talk about prime and irreducible elements and abstract our abstraction of the integers).
Ok, goodnight for now. Wish me luck (or you'll have to bear with Ring Theory in my Xanga for another semester).
