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.