= Ore condition
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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 $u a=b s$.
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.
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