  We introduce Backstr\"om pairs and Backstr\"om rings, study their derived
categories and construct for them a sort of categorical resolutions. For the
  latter we define the global dimension, construct a sort of semi-orthogonal
decomposition of the derived category and deduce that the derived dimension of
a Backstr\"om ring is at most $2$.
  Using this semi-orthogonal decomposition, we define a description of the
module category as the category of elements of a special bimodule.
  We also construct a partial tilting for a Backstr\"om pair to a ring of
triangular matrices and define the global dimension of the latter.
