  We study certain Schur functors which preserve singularity categories of
rings and we apply them to study the singularity category of triangular matrix
rings. In particular, combining these results with Buchweitz-Happel's theorem,
we can describe singularity categories of certain non-Gorenstein rings via the
stable category of maximal Cohen-Macaulay modules. Three concrete examples of
finite-dimensional algebras with the same singularity category are discussed.
