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?
2026-03-27 10:08:27.1774606107
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Requirements on ring for injective-projectiveness
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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
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Any semisimple artinian Ring, will do the trick infact such a ring may be characterised as: Every R-Module is both projective and injective.