Weak Dimension of FP-injective Modules Over Chain Rings

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.

Key Words:

• Chain ring
• FP-Injective module
• Injective module
• Weak dimension

2010 Mathematics Subject Classification:

• 13F30
• 13C11
• 13E05

Notes

We prefer “chain ring” to “valuation ring” to avoid confusion with “Manis valuation ring.”

Communicated by S. Bazzoni.