= Ore localization
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An <Ore localization> of a <noncommutative ring> $R$ at a <right denominator set> $S$ is a ring $RS^{-1}$ in which each element has the form $rs^{-1}$. Its canonical map sends every member of $S$ to a <unit> and is universal among <ring homomorphisms> doing so. The <right Ore condition> provides common denominators, and denominator reversibility makes fraction equality and multiplication well-defined.
The kernel is $\{r\in R:rs=0\text{ for some }s\in S\}$. Thus the map is injective when every denominator is a <regular element of a ring>, but it can kill elements when zero divisors are inverted. In a <right Noetherian domain>, localizing at all nonzero elements produces a <division ring>, the noncommutative counterpart of a <field of fractions>. A <left Ore fraction construction> uses denominators on the other side.
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