Relative singularity categories and singular equivalences

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