# Requirements on ring for injective-projectiveness

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?

• Feb 20 '14 at 16:10

Yes, the necessary and sufficient conditions are well-known.

The following conditions are equivalent for a ring $R$:

1. All injective right $R$ modules are projectve
2. All projective right $R$ modules are injective
3. The injective and projective right $R$ modules coincide
4. All of the above 3 conditions with "right" replaced with "left"
5. $R$ Noetherian on a side and self-injective on a side
6. $R$ is Artinian on both sides and self-injective on both sides
7. $R$ is quasi-Frobenius

Any semisimple artinian Ring, will do the trick infact such a ring may be characterised as: Every R-Module is both projective and injective.

• Just to be clear, semisimple artinian rings are characterized this way, yes, but not rings for which "injectives are projectives." Feb 20 '14 at 20:42