  The total right ring of quotients $Q_{\mathrm{tot}}^r(R),$ sometimes also
called the maximal flat epimorphic right ring of quotients or right flat
epimorphic hull, is usually obtained as a directed union of a certain family of
extension of the base ring $R$. In [K. Morita, Flat modules, injective modules
and quotient rings, Math. Z. 120 (1971) 25--40], $Q_{\mathrm{tot}}^r(R)$ is
constructed in a different way, by transfinite induction on ordinals. Starting
with the maximal right ring of quotients $Q_{\mathrm{max}}^r(R)$, its subrings
are constructed until $Q_{\mathrm{tot}}^r(R)$ is obtained.
  Here, we prove that Morita's construction of $Q_{\mathrm{tot}}^r(R)$ can be
simplified for rings satisfying condition (C) that every subring of the maximal
right ring of quotients $Q^r_{\mathrm{max}}(R)$ containing $R$ is flat as a
left $R$-module. We illustrate the usefulness of this simplification by
considering the class of right semihereditary rings all of which satisfy
condition (C). We prove that the construction stops after just one step and we
obtain a simple description of $Q^r_{\mathrm{tot}}(R)$ in this case. Lastly, we
study conditions that imply that Morita's construction ends in countably many
steps.

Rings and Algebras