The total right ring of quotients Qtotr(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], Qtotr(R)
Nearby in the stack
is constructed in a different way, by transfinite induction on ordinals. Starting with the maximal right ring of quotients
Qmaxr(R)
, its subrings are constructed until
Qtotr(R)
is obtained. Here, we prove that Morita's construction of
Qtotr(R)
can be simplified for rings satisfying condition (C) that every subring of the maximal right ring of quotients
Qmaxr(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
Qtotr(R)
in this case. Lastly, we study conditions that imply that Morita's construction ends in countably many steps.