Ore condition

Summary

The Ore condition is a condition on a multiplicative set $S$ of a noncommutative ring. It allows one to form fractions $a s^{-1}$ with denominators in $S$. It asks that a "left fraction" $s^{-1} a$ can always be rewritten as a "right fraction" $b t^{-1}$. Together with a mild cancellation condition, it is exactly what is needed for the Ore localization to exist. For commutative rings it always holds, so it explains why ordinary localization needs no extra hypothesis.