A note on the localization of generalized injective modules

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 some counter-examples to deny some questions proposed in \cite{B24}.