10 - Rings and Modules of Fractions

Summary

In the study of commutative rings, one meets rings of fractions early on, first as quotient fields of integral domains but later as a general construction method. Given a commutative ring R and a subset X of R, we want to find a “larger” ring in which the elements of X become units. First of all, since all products of elements of X would necessarily become units in the new ring, we may enlarge X and assume that X is multiplicatively closed and that 1 ∈ X. We then build a new ring RX⁻¹ (often written R_X in the commutative literature, but this notation can cause confusion later) as a set of fractions r/x, where r ∈ R and x ∈ X. There must be an equivalence relation on these fractions, and the situation is made slightly more complex by the fact that X may contain zero-divisors, in which case the map R → RX⁻¹ taking r to r/1 is not injective. The correct equivalence relation turns out to be the following: We say that r/x and r′/x′ define the same element of RX⁻¹ if and only if (rx′ − r′x)y = 0 for some y ∈ X. Some easy calculations show that we can define a ring RX⁻¹ in this way.