Some remarks on injective envelopes on ring extensions

Let $f:S\to R$ be a ring extension. We introduce and study the properties of $(R,S{)}_{\star }$-injective modules and the existences of $(R,S{)}_{\star }$-injective envelopes. Besides, we show that every $R$-module has an $(R,S)$-injective envelope when $S$ is a pure-semisimple ring.