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  • Print publication year: 2004
  • Online publication date: November 2010

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 RX in the commutative literature, but this notation can cause confusion later) as a set of fractions r/x, where rR and xX. 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 RRX−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 yX. Some easy calculations show that we can define a ring RX−1 in this way.