Local Rings

Sorry for the break last Friday.

As long as we’re in the neighborhood — so to speak — we may as well define the concept of a “local ring”. This is a commutative ring which contains a unique maximal ideal. Equivalently, it’s one in which the sum of any two noninvertible elements is again noninvertible.

Why are these conditions equivalent? Well, if we have noninvertible elements and with invertible, then these elements generate principal ideals and . If we add these two ideals, we must get the whole ring, for the sum contains , and so must contain , and thus the whole ring. Thus and cannot both be contained within the same maximal ideal, and thus we would have to have two distinct maximal ideals.

Conversely, if the sum of any two noninvertible elements is itself noninvertible, then the noninvertible elements form an ideal. And this ideal must be maximal, for if we throw in any other (invertible) element, it would suddenly contain the entire ring.

Why do we care? Well, it turns out that for any manifold and point the algebra of germs of functions at is a local ring. And in fact this is pretty much the reason for the name “local” ring: it is a ring of functions that’s completely localized to a single point.

To see that this is true, let’s consider which germs are invertible. I say that a germ represented by a function is invertible if and only if . Indeed, if , then is certainly not invertible. On the other hand, if , then continuity tells us that there is some neighborhood of where . Restricting to this neighborhood if necessary, we have a representative of the germ which never takes the value zero. And thus we can define a function for , which represents the multiplicative inverse to the germ of .

With this characterization of the invertible germs in hand, it should be clear that any two noninvertible germs represented by and must have . Thus , and the germ of is again noninvertible. Since the sum of any two noninvertible germs is itself noninvertible, the algebra of germs is local, and its unique maximal ideal consists of those functions which vanish at .

Incidentally, we once characterized maximal ideals as those for which the quotient is a field. So which field is it in this case? It’s not hard to see that — any germ is sent to its value at , which is just a real number.

Rings

Okay, I know I’ve been doing a lot more high-level stuff this week because of the thing, but it’s getting about time to break some new ground.

A ring is another very well-known kind of mathematical structure, and we’re going to build it from parts we already know about. First we start with an abelian group, writing this group operation as . Of course that means we have an identity element , and inverses (negatives).

To this base we’re going to add a semigroup structure. That is, we can also “multiply” elements of the ring by using the semigroup structure, and I’ll write this as we usually write multiplication in algebra. Often the semigroup will actually be a monoid — there will be an identity element . We call this a “ring with unit” or a “unital ring”. Some authors only ever use rings with units, and there are good cases to be made on each side.

Of course, it’s one thing to just have these two structures floating around. It’s another thing entirely to make them interact. So I’ll add one more rule to make them play nicely together:

This is the familiar distributive law from high school algebra.

Notice that I’m not assuming the multiplication in a ring to be invertible. In fact, a lot of interesting structure comes from elements that have no multiplicative inverse. I’m also not assuming that the multiplication is commutative. If it is, we say the ring is commutative.

The fundamental example of a ring is the integers . I’ll soon show its ring structure in my thread of posts directly about them. Actually, the integers have a lot of special properties we’ll talk about in more detail. The whole area of number theory basically grew out of studying this ring, and much of ring theory is an attempt to generalize those properties.