# Tag Archives: injective modules

## 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 […]

## Self Injective Integral Domains are Fields: Two Proofs

For finite commutative rings, integral domains are the same as fields. This isn't too surprising, because an integral domain $R$ is a ring such that for every nonzero $a\in R$ the $R$-module homomorphism $R\to R$ given by $r\mapsto ra$ is injective. Fields are those rings for which all these maps are surjective. But injective and […]