A note on the localizations of generalized injective modules

Abstract Let R be a ring and let S be 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 u - S -coherent rings. Besides, we give an example to show the difference between S -injective modules and u - S -injective modules, and some counter-examples to deny questions proposed in [J. Baeck, S -injective modules, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 118 2024, 1, Paper No. 20].