  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}.
