FGT-injective dimensions of Π-coherent rings and almost excellent extension
Y Xiang - Proceedings-Mathematical Sciences, 2010 - Springer
Y Xiang
Proceedings-Mathematical Sciences, 2010•SpringerWe study, in this article, the FGT-injective dimensions of Π-coherent rings. If R is right Π-
coherent, and TI (resp. TF) stands for the class of FGT-injective (resp. FGT-flat) R-modules
(n≥ 0), we show that the following are equivalent:(1) FGT-Id R (R)≤ n (2) If 0→ M→ F 0→ F
1→... is a right TF-resolution of left R-module M, then the sequence is exact at F k for k≥ n−
1 (3) For every flat right R-module F, there is an exact sequence 0→ F→ A 0→ A 1→...→ A
n→ 0 with each A i∈ TI (4) For every injective left R-module A, there is an exact sequence …
coherent, and TI (resp. TF) stands for the class of FGT-injective (resp. FGT-flat) R-modules
(n≥ 0), we show that the following are equivalent:(1) FGT-Id R (R)≤ n (2) If 0→ M→ F 0→ F
1→... is a right TF-resolution of left R-module M, then the sequence is exact at F k for k≥ n−
1 (3) For every flat right R-module F, there is an exact sequence 0→ F→ A 0→ A 1→...→ A
n→ 0 with each A i∈ TI (4) For every injective left R-module A, there is an exact sequence …
Abstract
We study, in this article, the FGT-injective dimensions of Π-coherent rings. If R is right Π-coherent, and TI (resp. TF) stands for the class of FGT-injective (resp. FGT-flat) R-modules (n ≥ 0), we show that the following are equivalent:
- (1) FGT - Id R (R) ≤ n
- (2) If 0 → M → F 0 → F 1 → ... is a right TF-resolution of left R-module M, then the sequence is exact at F k for k ≥ n − 1
- (3) For every flat right R-module F, there is an exact sequence 0 → F → A 0 → A 1 → ... → A n → 0 with each A i ∈ TI
- (4) For every injective left R-module A, there is an exact sequence 0 → F n → ... → F 1 → F 0 → A → 0 with each F i ∈ TF
- (5) If ... → I 1 → I 0 → M → 0 is a minimal left TI-resolution of a right R-module M, then the sequence is exact at I k for k ≥ n − 1.
Further, we characterize such homological dimension in terms of TI-syzygy and TF-cosyzygy of modules. Finally, we consider almost excellent extensions of rings. These extend the corresponding results in [10] as well.
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