Completions of rings and modules

So, we saw in the previous post that completion can be defined generally for abelian groups. Now, to specialize to rings and modules.

 Rings

 The case in which we are primarily interested comes from a ring with a descending filtration (satisfying ), which implies the are ideals; as we saw, the completion will also be a ring. Most often, there will be an ideal such that , i.e. the filtration is -adic. We have a completion functor from filtered rings to rings, sending . Given a filtered -module , there is a completion , which is also a -module; this gives a functor from filtered -modules to -modules.

 

How to tell if a ring is Noetherian

I briefly outlined the definition and first properties of Noetherian rings and modules a while back.  There are several useful and well-known criteria to tell whether a ring is Noetherian, as I will discuss in this post.  Actually, I’ll only get to the first few basic ones here, though these alone give us a lot of tools for, say, algebraic geometry, when we want to show our schemes are relatively well-behaved.  But there are plenty more to go.

 Hilbert’s basis theorem

 It is the following: 

Theorem 1 (Hilbert) Let be a Noetherian ring. Then the polynomial ring is also Noetherian.

 

 

Why simple modules are often finite-dimensional, I

Today I want to talk (partially) about a general fact, that first came up as a side remark in the context of my project, and which Dustin Clausen, David Speyer, and I worked out a few days ago.  It was a useful bit of algebra for me to think about.

Theorem 1 Let be an associative algebra with identity over an algebraically closed field ; suppose the center is a finitely generated ring over , and is a finitely generated -module. Then: all simple -modules are finite-dimensional -vector spaces.

We’ll get to this after discussing a few other facts about rings, interesting in their own right.