Ore condition
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In mathematics, the Ore condition in ring theory is a condition introduced by Øystein Ore, in connection with the question of extending beyond commutative rings the construction of a field of fractions, or more general localization of a ring. The left Ore condition reads that for a ring R, and any pair a, b of non-zero elements, the sets aR and bR should intersect in more than the element 0. Symmetrically for the right Ore condition.
If R can be considered a subring of a division ring D then the Ore conditions must hold. Conversely, if R is a domain that satisfies the Ore condition, then R has a ring of fractions in the sense of a division ring D in which R embeds, and which is generated by the non-zero elements of R and their inverses.
A different, stronger version of the Ore conditions is usually given for the case where R is not a domain, namely that there should be a common multiple
- c = au = bv
with u, v not zero divisors. On this case Ore's theorem guarantees the existence of an over-ring called the (right or left) classical ring of quotients.
