Ore Condition - Multiplicative Sets

Multiplicative Sets

The Ore condition can be generalized to other multiplicative subsets, and is presented in textbook form in (Lam 1999, §10) and (Lam 2007, §10). A subset S of a ring R is called a right denominator set if it satisfies the following three conditions for every a,b in R, and s, t in S:

1. st in S; (The set S is multiplicatively closed.)
2. aS ∩ sR is not empty; (The set S is right permutable.)
3. If sa = 0, then there is some u in S with au = 0; (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, tor_S(M) = { m in M : ms = 0 for some s in S }, is an R-submodule isomorphic to Tor₁(M,RS−1), and the module M ⊗_R RS−1 is naturally isomorphic to a module MS−1 consisting of "fractions" as in the commutative case.