# 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 $R$-module is injective, then it turns out that every $R$-module is also projective. That’s just because every $R$-module being projective or injective is just another way of saying that every short exact sequence of $R$-modules splits.

What about if we weaken the Baer criterion? Let’s say that an $R$-module $M$ is called p-injective if the Baer criterion holds for principal ideals. That is, $M$ is p-injective if for every principal left ideal $I$ of $R$, every map $I\to M$ extends to a map $R\to M$.

Every injective $R$-module is also p-injective, but the converse is not true. Indeed: it was R. Yue Chi Ming that noticed that a ring $R$ has every module p-injective if and only if $R$ is von Neumann regular. I’ve talked about this rings before because they’re cool.

Briefly, a ring $R$ is called von Neumann regular if for each $a\in R$ there exists an $x\in R$ such that $axa = a$. Let’s suppose we have one of these von Neumann regular rings $R$. Let $M$ be an $R$-module. We claim it’s p-injective. Indeed, suppose $g:Rb\to M$ is a map from a principal left ideal $Rb$ to $M$. Choose an $x\in R$ such that $bxb = b$. Such an $x$ exists by definition. Then $G:R\to M$ defined by $G(r) = rg(xb)$ is an extension of $g$. Therefore, $M$ is p-injective.

Therefore, every $R$-module is p-injective if $R$ is von Neumann regular. The converse is similar, and I’ll leave it as an exercise.

Let’s take an example: an infinite product of fields. Such a ring is easily verified to be von Neumann regular. Therefore, every module is p-injective. But such a ring is not semisimple, and therefore some of its modules are not injective.