What is this page? It's the start of a new project to compile as many counterexamples in algebra as possible. It is a work in progress, and if you would like to help write up a quick example, please send it to me as a guest post following roughly the outline of this post.

## Group Theory

- A finitely-presented group that is not Hopfian (and hence not residually finite)

## Module Theory

- Projective modules that are not free (nine examples!)
- A finitely-generated flat module that is not projective

## Ring Theory

- A non-Noetherian domain in which finitely generated ideals are principal
- A Cohen-Macaulay ring that is not regular
- A Noetherian and Artinian ring that is not semisimple
- A non-Noetherian subring of a Noetherian ring
- A subalgebra of a matrix algebra that is not separable
- A zero-dimensional ring that is not von Neumann regular