Let \(R\) be an associative ring with unit. The notion of a duality pair was introduced by J. Gillespie in [J. Pure Appl. Algebra 223, No. 8, 3425–3435 (2019; Zbl 1409.13034)] as a pair, \((_R{\mathcal L},{\mathcal A}_R)\), of classes of \(R\)-modules, left \(R\)-modules the first and right \(R\)-modules the second, satisfying that \(L\in {\mathcal L}\) if and only if \(L^+ =Hom_{\mathbb Z}(M,{\mathbb Q}/{\mathbb Z}) \in {\mathcal A}\) and that \(\mathcal A\) is closed under direct summands and finite direct sums. By using these pairs, a relative homological algebra by defining Gorenstein \((\mathcal L, \mathcal A)\)- projective, injective and flat was developed.
In this paper, for some particular duality pairs it is characterized when these new notions coincide with the classical notions of projective, injective and flat modules. This result is applied to obtain a characterization of strongly left CM-free, i.e. every Gorenstein projective left \(R\)-module is projective, to compare the global dimension of the ring with the relative global dimension respect to \((\mathcal L, \mathcal A)\), and to give a sufficient condition for the relative derived category with respect to Gorenstein \((\mathcal L, \mathcal A)\)-projective to be compactly generated