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.

One thought on “Categories? Rings?

  1. good one! it will be nice if you expose the reader to more concrete prime ideals and demonstrate its natural setting…

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