Total ring of fractions

In abstract algebra, the total ring of fractions is a construction that generalizes the concept of localization from integral domains to more general rings. Specifically, it provides a way to create a new ring that contains the original ring and allows for division by certain elements, including non-zero divisors. ### Definition: Given a ring \( R \) (not necessarily an integral domain) and a set \( S \) of elements in \( R \) that contains the non-zero divisors (i.e.