Ore Condition - General Idea

General Idea

The goal is to construct the right ring of fractions R 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. 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.