Rad-supplements in injective modules

We introduce and study the notion of $\mathrm{Rad}$-s-injective modules (i.e. modules which are $\mathrm{Rad}$-supplements in their injective hulls). We compare this notion with another generalization of injective modules. We show that the class of $\mathrm{Rad}$-s-injective modules is closed under finite direct sums. We characterize $\mathrm{Rad}$-s-injective modules over several type of rings, including semilocal rings, left hereditary rings and left Harada rings.