Construct a faithful injective module over the group algebra $KG$ where $K$ is a field and $G$ is arbitrary group.

Question

I can proceed a bit only when $K$ has characteristic $0$ and $|G| < \infty$. Since in this case, $KG$ is a finite dimensional semisimple algebra and any projective module is injective module. But I'm not sure how to systematically construct such faithful modules.

Any help or comments are welcomed!

Answer 1

One faithful injective left module over an arbitrary unital ring $R$ over a field $K$ is the module $N = \operatorname{Hom}_K(R,R)$ with the left action defined by $(r\phi)(s) = \phi(sr)$. $N$ is faithful because $R$ embeds into $N$ via right-multiplication. To show $N$ is injective, show the functors $\operatorname{Hom}_R(-,N)$ and $\operatorname{Hom}_K(-,R)$ are isomorphic: $R$ is an injective $K$-module (because any $K$-module is injective), so $N$ is an injective $R$-module.

Answer 2

It's known that $R[G]$ is right self-injective exactly when $R$ is right self-injective and $G$ is finite.

So if you pick any right self-injective ring $R$ (a field $K$ being included) then $R[G]$ itself is faithful and injective as a right module over itself.

If you want to do it for arbitrary $G$, you can just take the injective envelope of $R[G]$, which will be automatically faithful (since it contains a copy of $R[G]$), and injective by design.