nLab ring of fractions




Constructivism, Realizability, Computability



In the same vein that commutative rings are to integral domains and GCD rings are to GCD domains, rings of fractions are to field of fractions.


Let RR be a commutative ring. A multiplicative submonoid of RR is a subset SRS \subseteq R with an injection m:SRm:S \hookrightarrow R, an element 1 SS1_S \in S and a function () S():S×SS(-)\cdot_S(-):S \times S \to S, such that m(1 S)=1m(1_S) = 1 and for all elements aSa \in S and bSb \in S, s(a Sb)=s(a)s(b)s(a \cdot_S b) = s(a) \cdot s(b). The localization of RR away from a multiplicative submonoid SS is the initial commutative ring R[S 1]R[S^{-1}] with a commutative ring homomorphism h:RR[S 1]h:R \to R[S^{-1}] and a function () 1:SR[S 1](-)^{-1}:S \to R[S^{-1}], such that for all aSa \in S, h(m(a))a 1=1h(m(a)) \cdot a^{-1} = 1.

The ring of fractions R[Reg(R) 1]R[\mathrm{Reg}(R)^{-1}] is the localization of RR away from the multiplicative subset of regular elements Reg(R)\mathrm{Reg}(R). Its elements are fractions ab\frac{a}{b} where aRa\in R and bReg(R)b\in \mathrm{Reg}(R) which are by the definition the equivalence classes of pairs (a,b)R×Reg(R)(a,b) \in R\times \mathrm{Reg}(R) and (a,b)(c,d)(a,b)\sim (c,d) iff ad=bca \cdot d = b \cdot c. The addition is given by the formula

ab+cd=ad+bcbd\frac{a}{b}+\frac{c}{d} = \frac{a \cdot d+b \cdot c}{b \cdot d}

and multiplication by

abcd=acbd\frac{a}{b} \cdot \frac{c}{d} = \frac{a \cdot c}{b \cdot d}

For aRa \in R, bReg(R)b \in \mathrm{Reg}(R), cReg(R)c \in \mathrm{Reg}(R), and dReg(R)d \in \mathrm{Reg}(R), division is given by

ab÷cd=adbc\frac{a}{b} \div \frac{c}{d} = \frac{a \cdot d}{b \cdot c}



Last revised on December 9, 2022 at 00:29:01. See the history of this page for a list of all contributions to it.