a, b in R, and s, t in S: st in S; (The set S is multiplicatively closed.) aS ∩ sR is not empty; (The set S is right permutable.) If , then there is some u in S with ; (The set S is right reversible.) If S is a right denominator set, then one can construct the ring of right fractions RS−1 similarly to the commutative case. If S is taken to be the set of regular elements (those elements a in R such that if b in R is nonzero, then ab and ba are nonzero), then the right Ore condition is simply the requirement that S be a right denominator set. Many properties of commutative localization hold in this more general setting. If S is a right denominator set for a ring R, then the left R-module RS−1 is flat. Furthermore, if M is a right R-module, then the S-torsion, is an R-submodule isomorphic to , and the module is naturally isomorphic to a module MS−1 consisting of "fractions" as in the commutative case. Notes References External links PlanetMath page on Ore condition PlanetMath page on Ore's theorem PlanetMath page on classical ring of quotients Ring theory