Some Remarks on Gorenstein Projective Precovers

The existence of the Gorenstein projective precovers over arbitrary rings is an open question. In this paper, we make use of three diferent techniques addressing intrinsic and homological properties of several classes of relative Gorenstein projective $R$-modules, among them including the Gorenstein projectives and Ding projectives, with the purpose of giving some situations where Gorenstein projective precovers exists. Within the development of such techniques we obtaint a family of hereditary and complete cotorsion pairs and hereditary Hovey triples that comes from relative Gorenstein projective $R$-modules. We also study a class of Gorenstein projective $R$-modules relative to the Auslander class $\mathcal{A}_C(R)$ of a semidualizing $(R,S)$-bimodule $_R C _S$, where we make use of a property of "reduction".