Classical ring of quotients

In mathematics, especially in the area of algebra known as ring theory, the Ore condition 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 generally localization of a ring. The right Ore condition for a multiplicative subset S of a ring R is that for a ∈ R and s ∈ S, the intersection aS ∩ sR ≠ ∅. A (non-commutative) integral domain for which the set of non-zero elements satisfies the right Ore condition is called a right Ore domain. The left case is defined similarly.

The goal is to construct the right ring of fractions R[S⁻¹] with respect to multiplicative subset S. In other words, we want to work with elements of the form as⁻¹ and have a ring structure on the set R[S⁻¹]. The problem is that there is no obvious interpretation of the product (as⁻¹)(bt⁻¹); indeed, we need a method to "move" s⁻¹ past b. This means that we need to be able to rewrite s⁻¹b as a product b₁s₁⁻¹. Suppose s⁻¹b = b₁s₁⁻¹ then multiplying on the left by s and on the right by s₁, we get bs₁ = sb₁. Hence we see the necessity, for a given a and s, of the existence of a₁ and s₁ with s₁ ≠ 0 and such that as₁ = sa₁.

Since it is well known that each integral domain is a subring of a field of fractions (via an embedding) in such a way that every element is of the form rs⁻¹ with s nonzero, it is natural to ask if the same construction can take a noncommutative domain and associate a division ring (a noncommutative field) with the same property. It turns out that the answer is sometimes "no", that is, there are domains which do not have an analogous "right division ring of fractions".

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