Category Archives: modules

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 […]