Counterexamples In Algebra

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

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

Module Theory

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

Ring Theory

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