symmetric monoidal (∞,1)-category of spectra
constructive mathematics, realizability, computability
propositions as types, proofs as programs, computational trinitarianism
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 $R$ be a commutative ring. A multiplicative submonoid of $R$ is a subset $S \subseteq R$ with an injection $m:S \hookrightarrow R$, an element $1_S \in S$ and a function $(-)\cdot_S(-):S \times S \to S$, such that $m(1_S) = 1$ and for all elements $a \in S$ and $b \in S$, $s(a \cdot_S b) = s(a) \cdot s(b)$. The localization of $R$ away from a multiplicative submonoid $S$ is the initial commutative ring $R[S^{-1}]$ with a commutative ring homomorphism $h:R \to R[S^{-1}]$ and a function $(-)^{-1}:S \to R[S^{-1}]$, such that for all $a \in S$, $h(m(a)) \cdot a^{-1} = 1$.
The ring of fractions $R[\mathrm{Reg}(R)^{-1}]$ is the localization of $R$ away from the multiplicative subset of regular elements $\mathrm{Reg}(R)$. Its elements are fractions $\frac{a}{b}$ where $a\in R$ and $b\in \mathrm{Reg}(R)$ which are by the definition the equivalence classes of pairs $(a,b) \in R\times \mathrm{Reg}(R)$ and $(a,b)\sim (c,d)$ iff $a \cdot d = b \cdot c$. The addition is given by the formula
and multiplication by
For $a \in R$, $b \in \mathrm{Reg}(R)$, $c \in \mathrm{Reg}(R)$, and $d \in \mathrm{Reg}(R)$, division is given by
The ring of fractions of the integers is the rational numbers $\mathbb{Q} \coloneqq \mathbb{Z}[\mathrm{Reg}(\mathbb{Z})^{-1}]$.
The ring of fractions of a Heyting integral domain is a Heyting field.
The ring of fractions of a strict approximate integral domain is a local ring.
The ring of fractions of any commutative ring is a prefield ring.
Last revised on December 9, 2022 at 00:29:01. See the history of this page for a list of all contributions to it.