Past exam of the non-commutative rings course of the University of Oxford / 2023 / 2 / a / i / Solution

A multiplicative subset $S$ is a left Ore set if for every $s\in S$ and $r\in R$ there are $s^{\prime }\in S$ and $r^{\prime }\in R$ withEquivalently, the subsets $Sr$ and $Rs$ intersect for each such pair. This is the common-left-multiple condition needed for Ore localization with denominators on the left. In a noncommutative domain, no extra condition concerning annihilation by denominators is needed, since all members of $S$ are regular elements of a ring.