Annihilators and decompositions of singularity categories

Given any commutative Noetherian ring R and an element x in R , we consider the full subcategory C(x) of its singularity category consisting of objects for which the morphism that is given by the multiplication by x is zero. Our main observation is that we can establish a relation between C(x), C(y) and C(xy) for any two ring elements x and y . Utilizing this observation, we obtain a decomposition of the singularity category and consequently an upper bound on the dimension of the singularity category.