What requirements could be asked (minimal) of a ring R, so that any module M on R which is injective must also be projective? Is this possible?
Yes, the necessary and sufficient conditions are well-known.
The following conditions are equivalent for a ring $R$:
- All injective right $R$ modules are projectve
- All projective right $R$ modules are injective
- The injective and projective right $R$ modules coincide
- All of the above 3 conditions with "right" replaced with "left"
- $R$ Noetherian on a side and self-injective on a side
- $R$ is Artinian on both sides and self-injective on both sides
- $R$ is quasi-Frobenius