Strongly $FP$-injective dimensions and Gorenstein projective precovers

Abstract: The existence of the Gorenstein projective precovers over arbitrary rings is an open question. In this paper, we use the strongly $FP$-injective modules to present some situations where there exists Gorenstein projective precovers, also make use of a semidualizing bimodule $_S C_R$ for the same purpose. We prove that the pair $(\mathcal{GP} , \mathcal{GP} ^{\perp})$ forms a complete cotorsion pair under some mild conditions and from there we provide a family of induced cotorsion pairs. We also prove that if $M$ has finite strongly $FP$-injective dimension, then the Gorenstein projective (resp. Gorenstein flat) dimension and the projective (resp. flat) dimension coincide over a general ring