Category Archives: homological-algebra

## 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 […]

## Dimension zero rings for three types of dimension

There are all sorts of notions of dimension that can be applied to rings. Whatever notion you use though, the ones with dimension zero are usually fairly simple compared with the rings of higher dimension. Here we'll look at three types of dimension and state what the rings of zero dimension look like with respect […]