Classical ring of quotients

Remarks

The right classical ring of quotients is the Ore localization $R C^{-1}$. It exists if and only if $C$ satisfies the right Ore condition; reversibility holds automatically, since elements of $C$ are not zero divisors. It is then unique up to an isomorphism that fixes $R$.

Example: for a commutative integral domain $R$, the set $C$ is $R \setminus \set{0}$, and the classical ring of quotients is the field of fractions. For $R = \Z$ it is $\Q$: each nonzero integer is invertible in $\Q$, and each rational number is $a c^{-1}$ with $a, c \in \Z$ and $c \ne 0$.

Non-example: the free algebra $k\langle x, y \rangle$ over a field $k$ is a domain, so $C$ consists of all nonzero elements. It has no right classical ring of quotients, because $C$ fails the right Ore condition: $x C \cap y R = \emptyset$.

A ring $R$ with a right classical ring of quotients $Q$ is called a right order in $Q$. A right Ore domain has a right classical ring of quotients that is a division ring.