  Various generalizations of the concept of injectivy, in particular injectivy
with respect to a specific class of morphisms, have been intensively studied
throughout the years in different categories. One of the important kinds of
injectivy studied in the category R-Mod of R-modules is ${\tau}$-injectivy, for
a torsion theory ${\tau}$, or in the other words r-injectivy, where r is the
induced idempotent radical by ${\tau}$. In this paper, we introduce the notion
of r-injectivy, for a Hoehnke radical r in the category S-Act of S-acts and we
study the main properties of this kind of injectivy. Indeed, we show that this
kind of injectivy is well behavior and also we present a Bear Theorem for
r-injective S-acts. We then consider r-injectivy for a Kurosh-Amitsur radical r
and we give stronger results in this case. Finally we present conditions under
which r-injective S-acts are exactly injective ones and we give a
characterization for injective S-acts
