Generalized coherent domains of self-weak injective dimension at most one

Abstract

Let R be a commutative ring with identity. An R-module E is said to be weak injective if ${\text{Ext}}_R^1(N,E)=0$ for any super finitely presented R-module N. In this article, for a domain R with quotient field $K\ne R$, it is characterized when K/R as an R-module is weak injective. Also, a couple of characterizations of generalized coherent domains of self-weak injective dimension one are given. As corollaries, we characterize Gorenstein Dedekind (resp., Gorenstein Prüfer) domains.

Keywords:

• Generalized coherent ring
• Gorenstein projective module
• super finitely presented module
• 1-GFC-domain

2010 Mathematics Subject Classification:

• 13D05
• 13C11
• 13G05

Acknowledgements

We would like to thank the referee for her/his valuable comments which improved the original version of this manuscript. This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2017R1D1A3B03033342).