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Let R be a ring and S a multiplicative subset of R. In this note, we study the localization of S-injective modules and u-S-injective modules under S-Noetherian rings and u-S-Noetherian rings, respectively. The u-S-absolutely pure property is showed to be preserved under localizations over S-coherent rings. Besides, we give an example to show the difference between S-injective modules and u-S-injective modules, and an example to deny 2, Question 3. 11.
Xiaolei Zhang (Sat,) studied this question.