Consider a ring $R$ and its module category $R-\mathrm{Mod}$. In a full exact subcategory $\cal{C}$ (whose exact sequences are exact sequences in $R-\mathrm{Mod}$), it is not necessary that an injective object in $\cal{C}$ is injective as an $R$-module.
My question is: Under what condition, the injective objects in $\cal{C}$ are injective $R$-modules?
Known example: $\cal{C}=$ cat of f.g. $R$-modules. But I want a more general criterion.