For any ring $R$, the $R$-module $Hom_{\mathbb Z}(R,\mathbb Q/\mathbb Z)$ is an injective cogenerator of the category of $R$-modules. Here $\mathbb Q/\mathbb Z$ can be replaced with $\mathbb R/\mathbb Z$ or any other injective cogenerator of the category of abelian groups. When $R$ is an algebra over a field $\mathbb K$, another example of an injective cogenerator of the category of $R$-modules is $Hom_{\mathbb K}(R,\mathbb K)$.
For any ring $R$, the $R$-module $Hom_{\mathbb Z}(R,\mathbb Q/\mathbb Z)$ is an injective cogenerator of the category of $R$-modules.