The flat dimensions of injective modules
If \(M\) is a module, let \(\text{fd}(M)\), \(\text{pd}(M)\) and \(\text{id}(M)\) denote the flat dimension, projective dimension, and injective dimension of \(M\), respectively. Let \(R\) be an associative ring with identity and \(\text{Mod-}R\) (\(R\text{-Mod})\) be the category of unitary right (left) \(R\)-modules. The authors prove, among other things, that for any ring \(R\) we have, \(l.\text{FWD}(R) \leq r.\text{IFD}(R) \leq \text{WD}(R)\), where \(\text{WD}(R)\) is the weak dimension of the ring \(R\), \(l.\text{FWD}(R) = \sup\{\text{fd}(M) \mid M \in R\text{-Mod}\) with \(\text{fd}(M) < \infty\}\) and \(r.\text{IFD}(R) = \sup\{\text{fd}(E)\mid E\in \text{Mod- }R\) with \(E\) injective\}. Let \(M^ + = \text{Hom}_ \mathbb{Z}(M,\mathbb{Q}/\mathbb{Z})\) for an \(R\)-module. The authors also prove that the following are identical for any ring \(R\): (1) \(r.\text{IFD}(R)\), (2) \(\sup\{\text{fd}(M) \mid M \in \text{Mod-}R\) with \(M\) FP-injective\}, (3) \(\sup\{\text{fd}(M) \mid M \in \text{Mod-}R\) with \(\text{FP-id}(M) < \infty\}\), and (4) \(\text{sup}\{\text{fd}(M^ +) \mid M \in R\text{- Mod}\) with \(\text{fd}(M) < \infty\}\).
scientific article; zbMATH DE number 492770
scientific article