A conventional wisdom of building $\mathbb{Q}$ from $\mathbb{Z}$ is the construction of the Field of Quotients of the Integral Domain. Are there alternatives?
For example, can the $\mathbb{Q}$ be described as a Quotient Ring:
$\mathbb{Z}[x,y]/\langle xy-1 \rangle$? I struggle to exhibit an isomorphism of this ring into $\mathbb{Q}$.
Reflecting onto the standard construction of the Field of Quotients
$\mathbb{Z} \oplus \mathbb{Z} / \{ (a,b) \sim (c,d) \mid ad = bc \} $
it prompts that perhaps we need to construct a projective variety out of $\mathbb{Z} \oplus \mathbb{Z}$ (if there is such a thing), and work with homogeneous polynomials?
In general, given the 2 ring building operations: direct sum, and making a polynomial ring, is there a ring $R(\mathbb{Z})$, e.g.
$ (\mathbb{Z} \oplus \mathbb{Z})[x,y,z] \oplus (\mathbb{Z} \oplus \mathbb{Z} \oplus \mathbb{Z})[x,t] $
and an ideal $I \subset R(\mathbb{Z})$ such that $ R(\mathbb{Z})/I$ is isomorphic to $\mathbb{Q}$?
Best Answer
Yes, indeed, there are alternatives to the common pair-based (normal form) construction of rings of fractions. Below are a couple alternatives.
The fraction ring (localization) $\rm\,S^{-1} R\,$ is, conceptually, the universal way of adjoining inverses of $\rm\,S\,$ to $\rm\,R.\,$ The simplest way to construct it is $\rm\,S^{-1} R = R[x_i]/(s_i x_i - 1).\,$ This allows one to exploit the universal properties of quotient rings and polynomial rings to quickly construct and derive the basic properties of localizations (avoiding the many tedious verifications always "left for the reader" in the more commonly presented pair approach). For details of this folklore see e.g. the exposition in section 11.1 of Rotman's Advanced Modern Algebra, or Voloch, Rings of fractions the hard way, and see this answer for a simplification to both expositions, and generalizations (e.g. to noncommutative rings).
Note that pairs $(a,s)$ representing $a/s$ in the common pair-based approach are simply convenient "normal" forms of the polynomials used in the above presentation based approach (with the ring structure transported to the normal forms in standard way, e.g. see here).
Another approach - not as well-known - is via partial endomorphisms - see below, excerpted from Lambek's book Lectures on Rings and Modules.