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A model structure on a category is a formal way of introducing a homotopytheory on that category, and if the model structure is abelian and hereditary,its homotopy category is known to be triangulated. So a good way to both buildand model a triangulated category is to build a hereditary abelian modelstructure. Given a ring $R$ and a (non necessarily semidualizing) left$R$-module $C$, we introduce and study new concepts of relative Gorensteincotorsion and cotorsion modules: $\rm G_C$-cotorsion and (strongly)$\mathcal{C}_C$-cotorsion. As an application, we prove that there is a uniquehereditary abelian model structure on the category of left $R$-modules, inwhich the cofibrations are the monomorphisms with $\rm G_C$-flat cokernel andthe fibrations are the epimorphisms with $\mathcal{C}_C$-cotorsion kernelbelonging to the Bass class $\mathcal{B}_C(R)$. In the second part, when $C$ isa semidualizing $(R,S)$-bimodule, we investigate the existence of abelian modelstructures on the category of left (resp., right) $R$-modules where thecofibrations are the epimorphisms (resp., monomorphisms) with kernel (resp.,cokernel) belonging to the Bass (resp., Auslander) class $\mathcal{B}_C(R)$(resp., $\mathcal{A}_C(R)$). We also study the class of $\rm G_C$-flat modulesand the Bass class from the Auslander-Buchweitz approximation theory point ofview. We show that they are part of weak AB-contexts. As the concept of weakAB-context can be dualized, we also give dual results that involve the class of$\rm G_C$-cotorsion modules and the Auslander class.

Relative Gorenstein flat modules and Foxby classes and their model structures