Relative singularity categories and singular equivalences

Let R be a right noetherian ring. We introduce the concept of relative singularity category Delta chi(R) of R with respect to a contravariantly finite subcategory chi of mod- R. Along with some finiteness conditions on chi, we prove that Delta chi(R) is triangle equivalent to a subcategory of the homotopy category K-ac(chi) of exact complexes over chi. As an application, a new description of the classical singularity category D-sg(R) is given. The relative singularity categories are applied to lift a stable equivalence between two suitable subcategories of the module categories of two given right noetherian rings to get a singular equivalence between the rings. In different types of rings, including path rings, triangular matrix rings, trivial extension rings and tensor rings, we provide some consequences for their singularity categories.