ABSTRACT
The classical projective resolutions are known to exist over arbitrary rings. When it comes to the existence of the Gorenstein projective resolutions, though, this is still an open problem. Their existence over Gorenstein rings is known. This chapter shows that the class of Gorenstein projective modules is special precovering over any left GF-closed ring R such that every Gorenstein projective module is Gorenstein flat and every Gorenstein flat module has finite Gorenstein projective dimension. This class of rings includes that of right coherent and left n-perfect rings. The chapter shows that this inclusion is a strict one, and gives examples of left GF-closed and left n-perfect rings that have the desired properties (every Gorenstein projective is Gorenstein flat and every Gorenstein flat has finite Gorenstein projective dimension), and that are not right coherent. It shows that over a local n-Gorenstein ring R every finitely generated module has a finite right Gorenstein projective resolution.
