  In this paper, we study the notion of $\phi$-injectivity in the special case
that $\phi=0$. For an arbitrary locally compact group $G$, we characterize the
0-injectivity of $L^{1}(G)$ as a left $L^{1}(G)$ module. Also, we show that
$L^{1}(G)^{**}$ and $L^{p}(G)$ for $1<p<\infty$ are 0-injective Banach
$L^{1}(G)$ modules.
