A Note on Relatively Injective $C_0(S)$-Modules $C_0(S)$

In this note we discuss some necessary and some sufficient conditions for the relative injectivity of the $C_0(S)$-module $C_0(S)$, where $S$ is a locally compact Hausdorff space. We also give a Banach module version of Sobczyk's theorem. The main result of the paper is as follows: if the $C_0(S)$-module $C_0(S)$ is relatively injective, then $S=\beta(S\setminus \{s\})$ for any limit point $s\in S$.