A Simplification of Morita’s Construction of Total Right Rings of Quotients for a Class of Rings
نویسنده
چکیده
The total right ring of quotients Qtot(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 [16], Qtot(R) is constructed in a different way, by transfinite induction on ordinals. Starting with the maximal right ring of quotients Qmax(R), its subrings are constructed until Qtot(R) is obtained. Here, we prove that Morita’s construction of Qtot(R) can be simplified for rings satisfying condition (C) that every subring of the maximal right ring of quotients Qmax(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 Qtot(R) in this case. Lastly, we study conditions that imply that Morita’s construction ends in countably many steps.
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