Ore condition

The Ore condition is a common-multiple condition that allows fractions in a noncommutative ring. For a multiplicative set $S\subseteq R$ containing one and excluding zero, the right Ore condition requires that for every $a\in R$ and $s\in S$ there are $b\in R$ and $u\in S$ with $au=sb$. It lets a numerator pass a denominator while keeping denominators on the right. The left version instead asks for $ua=bs$.

If the denominators are regular elements of a ring, these conditions suffice for the corresponding Ore localization. For example, all nonzero elements of a commutative integral domain satisfy either condition, giving its field of fractions. In a right Noetherian domain, the nonzero elements satisfy the right condition and give a division ring. Without the condition, adjoining formal inverses need not retain the expected single-fraction form.