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Abstract
In this article, we study the weak global dimension of coherent rings in terms of the left FP-injective resolutions of modules. Let R be a left coherent ring and ℱ ℐ the class of all FP-injective left R-modules. It is shown that wD(R) ≤ n (n ≥ 1) if and only if every nth ℱ ℐ-syzygy of a left R-module is FP-injective; and wD(R) ≤ n (n ≥ 2) if and only if every (n − 2)th ℱ ℐ-syzygy in a minimal ℱ ℐ-resolution of a left R-module has an FP-injective cover with the unique mapping property. Some results for the weak global dimension of commutative coherent rings are also given.
ACKNOWLEDGMENTS
This research was partially supported by SRFDP (No. 20050284015), NSFC (No. 10331030), China Postdoctoral Science Foundation (No. 20060390926), Collegial Natural Science Research Program of Education Department of Jiangsu Province (No. 06KJB110033), and Jiangsu Planned Projects for Postdoctoral Research Funds (No. 0601021B). The authors would like to thank the referee for the helpful comments and suggestions.
Notes
Communicated by I. Swanson.