Abstract
We first prove that if a monomorphism is a phantom envelope of a left R-module, then its cokernel is pure-projective; if R is a left coherent ring and an epimorphism is an \(\mathrm{Ext}\)-phantom cover of a left R-module, then its kernel is pure-injective. Second, we characterize several rings such as left FC rings and left semihereditary rings using phantom envelopes and \(\mathrm{Ext}\)-phantom covers of modules.
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Acknowledgements
This research was supported by NSFC (11771202), NSF of Jiangsu Province of China (BK20160771) and Nanjing Institute of Technology of China (CKJA201707, JCYJ201842). The author wants to express his gratitude to the referee for the very helpful comments and suggestions.
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Mao, L. Phantom Envelopes and \(\mathrm{Ext}\)-Phantom Covers of Modules. Bull. Iran. Math. Soc. 46, 441–455 (2020). https://doi.org/10.1007/s41980-019-00268-6
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DOI: https://doi.org/10.1007/s41980-019-00268-6