(r,t)-injectivity in the category $S$-Act

In this paper, we show that injectivity with respect to the class $mathcal{D}$  of dense monomorphisms of an idempotent and weakly hereditary closure operator of an arbitrary category  well-behaves. Indeed, if $mathcal{M}$ is a subclass of monomorphisms, $mathcal{M}cap mathcal{D}$-injectivity  well-behaves. We also introduce the notion of $(r,t)$-injectivity in the category {bf S-Act}, where $r$ and $t$ are Hoehnke radicals, and discuss whether this kind of injectivity well-behaves.