Duality pairs, generalized Gorenstein modules, and Ding injective envelopes

Let R be a general ring. Duality pairs of R-modules were introduced by Holm-Jørgensen. Most examples satisfy further properties making them what we call semi-complete duality pairs in this paper. We attach a relative theory of Gorenstein homological algebra to any given semi-complete duality pair D = (L,A). This generalizes the homological theory of the AC-Gorenstein modules defined by Bravo–Gillespie–Hovey, and we apply this to other semi-complete duality pairs. The main application is that the Ding injective modules are the right side of a complete (perfect) cotorsion pair, over any ring. Completeness of the Gorenstein flat cotorsion pair over any ring arises from the same duality pair.