Preresolutions of noncommutative isolated singularities

We introduce the notion of right pre-resolutions (quasi-resolutions) for noncommutative isolated singularities, which is a weaker version quasi-resolutions introduced by Qin-Wang-Zhang. prove that noetherian bounded below and locally finite graded algebra with injective dimension 2 are always Morita equivalent. When we restrict to quadric hypersurfaces, hypersurface, singularity, admits pre-resolution. Besides, provide method verify whether hypersurface an singularity. An example hypersurfaces detailed computations indecomposable maximal Cohen-Macaulay modules included as well.