Shahid Beheshti University Categories and General Algebraic Structures with Applications 2345-5853 11 Special Issue Dedicated to Prof. George A. Grätzer 2019 07 01 (r,t)-injectivity in the category $S$-Act 169 196 76601 10.29252/cgasa.11.1.169 EN Mahdieh Haddadi Department of Mathematics, Faculty of Mathematics, Statistics and Computer Sciences, Semnan University, Semnan, Iran. Seyed Mojtaba Naser Sheykholislami Department of Mathematics, Faculty of Mathematics, Statistics and Computer Sciences, Semnan University, Semnan, Iran. Journal Article 2018 03 05 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. https://cgasa.sbu.ac.ir/article_76601_9c76323409aa064b4f265c828186eb91.pdf