Tag Archives: von neumann regular

Injective and p-injective

An $R$-module $M$ is called injective if the functor $\Hom_R(-,M)$ is exact. The well-known Baer criterion states that an $R$-module $M$ is injective if and only if for every ideal $I$ of $R$, every map $I\to M$ can actually be extended to a map $R\to M$. For example, $\Q$ is an injective $\Z$-module. If every […]

Local Rings of Weak Dimension Zero are Division

Let $R$ be a ring and $M$ be an $R$-module. The flat dimension of $M$ is the infimum over all lengths of flat resolutions of $M$. Usually, the flat dimension of $M$ is denoted by $\mathrm{fd}_R(M)$. For example, $\mathrm{fd}_{\mathbb{Z}}(\mathbb{Q}) = 0$. Since $\mathbb{Q}$ has projective dimension $1$, the flat dimension and projective dimension of a […]

Basic Examples of the Tensor Product and Flatness

The tensor product is one of the most important constructions in mathematics, and here we shall see my favourite examples of the tensor product in action, hopefully to illuminate its properties for beginners. Proofs or references are provided, but since the emphasis is on examples, the proofs that are given are terse and details are […]