Some remarks on injective envelopes on ring extensions

Let $f:S\rightarrow R$ be a ring extension. We introduce and study the properties of $(R, S)_\star $-injective modules and the existence of $(R, S)_\star $-injective envelopes. Moreover, we show that every $R$-module has an $(R, S)$-injective envelope when $S$ is a pure-semisimple ring, partly resolving a conjecture proposed by Guo [Comm. Algebra 52 (2024), 2868–2883].