On the Triviality of Gorenstein $$(\mathcal {L}, \mathcal {A})$$ -Modules

Let \((\mathcal {L}, \mathcal {A})\) be a bi-complete duality pair. We consider when the relative Gorenstein modules with respect to such a duality pair coincide with the classical homological modules. As applications, we characterize (weak) global dimension via such a duality pair, characterize strongly CM-freeness of flat-typed \((\mathcal {L}, \mathcal {A})\)-Gorenstein rings and obtain the compactly-generatedness of some relative derived categories. Applying the results into frequent duality pairs, our results unify the corresponding results for Gorenstein AC-modules and Ding modules.