Relative nonhomogeneous Koszul duality. Introduction

Let A be an associative ring and R be a subring in A. Derived Koszul duality is the functor Ext into R over A, or Tor with R over A, or Ext from R over A, enhanced to an equivalence of derived categories of modules.

The above definition raises many questions. To begin with, R is not an A-module. So what does this Ext and Tor notation even mean?