Raionals

Raionals are a mathematical construct that generalizes rational numbers through a process known as localization of rings. They appear in mathematical education and speculative frameworks to illustrate how fractions arise from algebraic structure.

Definition: Let R be a commutative ring with identity, and let S be a multiplicative subset of R that does not contain 0. The raionals of (R,S), denoted Ra(R,S), consist of symbols a/s with a in R and s in S, subject to the usual fraction relations: a/s = b/t if a t = b s in R. Addition and multiplication are defined by (a/s) + (b/t) = (at + bs)/(st) and (a/s)(b/t) = (ab)/(st). If R is an integral domain and S excludes zero divisors, Ra(R,S) forms a field; in particular, Ra(Z, Z\{0}) is isomorphic to the rational numbers Q.

Examples: Localization of Z at all nonzero integers gives Q; localization of Z at powers of 2 yields fractions with denominators a power of two. Localizing a polynomial ring F[x] at the nonzero polynomials with nonzero constant term yields the field of rational functions F(x).

Properties: The construction embeds R into Ra(R,S) and preserves many algebraic operations. When R is a field, Ra(R,S) is the same as R. The choice of S determines which denominators are allowed and can affect whether the raionals form a field, a domain, or a more general ring.

Applications and notes: Raionals provide a concise way to discuss fractions in an algebraic setting and connect to localization in commutative algebra. The term "raional" is nonstandard outside fictional or pedagogical use; standard terminology uses localization or the field of fractions.