Abstract
It is proven that the weak dimension of each FP-injective module over a chain ring which is either Archimedean or not semicoherent is less or equal to 2. This implies that the projective dimension of any countably generated FP-injective module over an Archimedean chain ring is less or equal to 3.
Notes
We prefer “chain ring” to “valuation ring” to avoid confusion with “Manis valuation ring.”
Communicated by S. Bazzoni.