Category Archives: ring-theory

The nonvanishing K_2(Z/4)

We saw previously that $K_2(F) = 0$ for a finite field $F$, where $K_2$ is the second $K$-group of $F$. It may be helpful to refer to that post for the definitions of this functor. I thought that it might be disappointing because we did all that work to compute the second $K$-group of a […]

The K-theory of finite fields: a synopsis

In my previous post, I proved that if $F$ is a finite field, then multiplicative group $F^\times$ is a cyclic group. This fact will play a small part in our description today of the $K$-theory of $F$. We will start by describing the classical $K$-theory of $F$ and then briefly talk about Quillen’s computation of […]

The multiplicative group of a finite field is cyclic

Today, we are going to prove that if $F$ is a finite field, the the multiplicative group $F^\times$ of $F$ is a cyclic group. Let’s start with an example of this phenomenon. Take $\F_5 = \{0,1,2,3,4\}$. Now, if we take powers of $2$ modulo $5$ we get $$2,2^2=4,2^3 = 3, 2^4 = 1.$$ Therefore, we […]

Roger Ming’s theorem on von Neumann regular rings

We say that an associative ring $A$ is von Neumann regular if for every $a\in A$ there exists a $x\in A$ such that $axa = a$. That is a rather strange condition, isn’t it? But, you can think of $x$ as a pseudoinverse to $a$. This weakening of inverses has a homological counterpart: if every […]

Exotic dimensions used in ring theory

Do you ever get the feeling that mathematics uses the word dimension a lot? Well, that’s for good reason. The concept of dimension is fundamental in mathematics. What is dimension? You can think of dimension as a numerical invariant characterizing the number of parameters required to do a certain thing. For example, for vector spaces, […]

Axiomatizability of classes of structures

Let’s talk about axiomatizability in first-order logic, with examples in ring theory. Let’s call a class $\Ccl$ of rings axiomatizable if there exists a set $T$ of first order sentences such that $C\in\Ccl$ if and only if $C$ is a model of $T$ (that is, satisfies every sentence in $T$.) What are some examples? The […]

Abelian categories: examples and nonexamples

I’ve been talking a little about abelian categories these days. That’s because I’ve been going over Weibel’s An Introduction to Homological Algebra. It’s a book I read before, and I still feel pretty confident about the material. This time, though, I think I’m going to explore a few different paths that I haven’t really given […]