Category Archives: modules

## A short survey of von Neumann regular rings

I've talked a lot about von Neumann regular rings on this blog, so I thought I'd write an informal short survey on them, collecting some facts we've already seen and many new ones. It should give you an idea of what von Neumann regular rings are. Most of the facts that I did not explicitly […]

## 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, […]

## Weak dimension one rings are axiomatizable

Let $R$ be a ring. In the previous post on pure exact sequences, we called an exact sequence $0\to A\to B\to C\to 0$ of left $R$-modules pure if its image under any functor $X\otimes -$ is an exact sequence of abelian groups for any right $R$-module $X$. Here is yet another characterization of purity: Theorem. […]

## A finitely generated flat module that is not projective

Let's see an example of a finitely-generated flat module that is not projective! What does this provide a counterexample to? If $R$ is a ring that is either right Noetherian or a local ring (that is, has a unique maximal right ideal or equivalently, a unique maximal left ideal), then every finitely-generated flat right $R$-module […]

## Stably free and the Eilenberg swindle

I already mentioned the idea of stably isomorphic for a ring $R$: two $R$-modules $A$ and $B$ are stably isomorphic if there exists a natural number $n$ such that $A\oplus R^n\cong B\oplus R^n$. Let's examine a specific case: if $A$ is stably isomorphic to a free module, then let's call it stably free. So, to […]

## Stable Isomorphisms, Grothendieck Groups: Example

If $a$ and $b$ are two real numbers and $ax = bx$, then we can't conclude that $a = b$ because $x$ may be zero. The same is true for tensor products of modules: if $A$ and $B$ are two left $R$-modules and $X$ is a right $R$-module, then an isomorphism $X\otimes_R A\cong A\otimes_R B$ […]

## When is a direct product of projective modules projective?

Over a field $k$, an arbitrary product of copies of $k$ is a free module. In other words, every vector space has a basis. In particular, this means that arbitrary products of projective $k$-modules are projective. Over the ring of integers, an arbitrary product of projective modules is not necessarily projective. In fact, a product […]

## First-order characterisations of free and flat…projective?

Here is an interesting question involving free, projective, and flat modules that I will leave to the readers of this blog for now. First, consider free modules. If $R$ is a ring, then every $R$-module is free if and only if $R$ is a division ring. The property of $R$ being a division ring can […]

## On a characterisation of Krull dimension zero rings

Here is one characterisation of commutative rings of Krull dimension zero: Theorem. A commutative ring $R$ has Krull dimension zero if and only if every element of the Jacobson radical ${\rm Jac}(R)$ of $R$ is nilpotent and the quotient ring $R/{\rm Jac}(R)$ is von Neumann regular. Recall that a ring $R$ is von Neumann regular […]