Tag Archives: counterexample

A zero-dimensional ring that is not von Neumann regular

An associative ring $R$ is called von Neumann regular if for each $x\in R$ there exists a $y\in R$ such that $x = xyx$. Now let $R$ be a commutative ring. Its dimension is the supremum over lengths of chains of prime ideals in $R$. So for example, fields are zero dimensional because the only […]

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 […]

Britton's lemma and a non-Hopfian fp group

In a recent post on residually finite groups, I talked a bit about Hopfian groups. A group $G$ is Hopfian if every surjective group homomorphism $G\to G$ is an isomorphism. This concept connected back to residually finite groups because if a group $G$ is residually finite and finitely generated, then it is Hopfian. A free […]

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 […]

When Are Discrete Subgroups Closed?

Let $ H\subseteq G$ be a subgroup of a topological group $ G$ (henceforth abbreviated "group"). If the induced topology on $ H$ is discrete, then we say that $ H$ is a discrete subgroup of $ G$. A commonplace example is the subgroup $ \mathbb{Z}\subseteq \mathbb{R}$: the integers are normal subgroup of the real […]